Unit IElectromagnetic field

Static Electric Fields

A changing magnetic field always produces an electric field, and conversely, a changing electric field always produces a magnetic field. This interaction of electric and magnetic forces gives rise to a condition in space known as an electromagnetic field. The characteristics of an electromagnetic field are expressed mathematically by Maxwell's equation. Vector A directed line segment. As such, vectors have magnitude and direction. Many physical quantities, for example, velocity, acceleration, and force, are vectors. Cross product the Cross Product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result. In many engineering and physics problems, it is handy to be able to construct a perpendicular vector from two existing vectors, and the cross product provides a means for doing so. The cross product is also known as the vector product, or Gibbs vector product. The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative). Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness". Certain features of the cross product can be generalized to other situations. For arbitrary choices of orientation, the cross product must be regarded not as a vector, but as a pseudovector. For arbitrary choices of metric, and in arbitrary dimensions, the cross product can be generalized by the exterior product of vectors, defining a two-form instead of a vector.

Fig 1.1 Illustration of the cross-product in respect to a right-handed coordinate system.

Fig 1.2 Finding the direction of the cross product by the right-hand rule. The cross product of two vectors a and b is denoted by a b. In a three-dimensional Euclidean space, with a usual right-handed coordinate system, it is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. The cross product is given by the formula

where is the measure of the angle between a and b (0 180), a and b are the magnitudes of vectors a and b, and is a unit vector perpendicular to the plane containing a and b. If the vectors a and b are collinear (i.e., the angle between them is either 0 or 180), by the above formula, the cross product of a and b is the zero vector 0. The direction of the vector is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector is coming out of the thumb (see the picture on the right). Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector is given by the left-hand rule and points in the opposite direction. Dot product The dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity.

where |a| and |b| denote the length (magnitude) of a and b is the angle between them. Since |a|cos() is the scalar projection of a onto b, the dot product can be understood geometrically as the product of this projection with the length of b. |a|cos() is the scalar projection of a onto b Coordinate System A coordinate system is a mathematical language that is used to describe geometrical objects analytically A cartesian coordinate system is one of the simplest and most useful systems of coordinates. It is constructed by choosing a point O designated as the origin. Through it three intersecting directed lines OX, OY, OZ, the coordinate axes, are constructed. The coordinates of a point P are x, the distance of P from the plane YOZ measured parallel to OX, and y and z, which are determined similarly (Fig. 1). Usually the three axes are taken to be mutually perpendicular, in which case the system is a rectangular cartesian one. Obviously a similar construction can be made in the plane, in which case a point has two coordinates (x,y).

fig 1.3 Cartesian coordinate system.

Cylindrical Coordinate System The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted h) which measures the height of a point above the plane. A point P is given as (r,,h). In terms of the Cartesian coordinate system:

r is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis. is the angle between the positive x-axis and the line OP', measured counterclockwise. h is the same as z. Thus, the conversion function f from cylindrical coordinates to Cartesian coordinates is .

Fig 1.4 :A point plotted with cylindrical coordinates Spherical Coordinates The three coordinates (, , ) are defined as:

0 is the distance from the origin to a given point P. 0 2 is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane. 0 is the angle between the positive z-axis and the line formed between the origin and P.

is referred to as the azimuth, while is referred to as the zenith, colatitude or polar angle.

and and lose significance when = 0 and loses significance when sin() = 0 (at = 0 and = 180). To plot a point from its spherical coordinates, go units from the origin along the positive z-axis, rotate about the y-axis in the direction of the positive x-axis and rotate about the z-axis in the direction of the positive y-axis. Coordinate system conversions As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others. Cartesian coordinate system The three spherical coordinates are obtained from Cartesian coordinates by:

Note that the arctangent must be defined suitably so as to take account of the correct quadrant of y / x. The atan2 or equivalent function accomplishes this for computational purposes. Conversely, Cartesian coordinates may be retrieved from spherical coordinates by:

Divergence of a Vector Field: In study of vector fields, directed line segments, also called flux lines or streamlines, represent field variations graphically. The intensity of the field is proportional to the density of lines. For example, the number of flux lines passing through a unit surface S normal to the vector measures the vector field strength.

Fig 1.5: Flux Lines We have already defined flux of a vector field as

....................................................(1.1) For a volume enclosed by a surface,

.........................................................................................(1.2) We define the divergence of a vector field at a point P as the net outward flux from a volume enclosing P, as the volume shrinks to zero.

.................................................................(1.3) Here is the volume that encloses P and S is the corresponding closed surface. . The flux

Let us consider a differential volume centered on point P(u,v,w) in a vector field through an elementary area normal to u is given by , ........................................(1.4)

Fig 1.6 Evaluation of divergence in curvilinear coordinate Net outward flux along u can be calculated considering the two elementary surfaces perpendicular to u .

.......................................(1.5) Considering the contribution from all six surfaces that enclose the volume, we can write

.......................................(1.6) Hence for the Cartesian, cylindrical and spherical polar coordinate system, the expressions for divergence can be written as: In Cartesian coordinates:

................................(1.7) In cylindrical coordinates:

....................................................................(1.8) and in spherical polar coordinates:

......................................(1.9) In connection with the divergence of a vector field, the following can be noted

Divergence of a vector field gives a scalar.

..............................................................................(1.10)

Divergence theorem :

Divergence theorem states that the volume integral of the divergence of vector field is equal to the net outward flux of the vector through the closed surface that bounds the volume. Mathematically, Proof: Let us consider a volume V enclosed by a surface S . Let us subdivide the volume in large number of cells. Let the kth cell has a volume and the corresponding surface is denoted by Sk. Interior to the volume, cells have common surfaces. Outward flux through these common surfaces from one cell becomes the inward flux for the neighboring cells. Therefore when the total flux from these cells are considered, we actually get the net outward flux through the surface surrounding the volume. Hence we can write:

......................................(1.11) In the limit, that is when written as . and the right hand of the expression can be

Hence we get

, which is the divergence theorem.

Curl of a vector field: We have defined the circulation of a vector field A around a closed path as .

Curl of a vector field is a measure of the vector field's tendency to rotate about a point. Curl , also written as is defined as a vector whose magnitude is maximum of the net circulation per unit area when the area tends to zero and its direction is the normal direction to the area when the area is oriented in such a way so as to make the circulation maximum. Therefore, we can write:

......................................(1.12) To derive the expression for curl in generalized curvilinear coordinate system, we first compute and to do so let us consider the figure 1.7:

Fig 1.7 Curl of a Vector If C1 represents the boundary of , then we can write ......................................(1.13) The integrals on the RHS can be evaluated as follows:

.................................(1.14)

................................................(1.15) The negative sign is because of the fact that the direction of traversal reverses. Similarly,

..................................................(1.16)

............................................................................(1.17)

Adding the contribution from all components, we can write:

........................................................................(1.18)

Therefore,

......................................................(1.19)

In the same manner if we compute for

and

we can write,

.......(1.20) This can be written as,

......................................................(1.21)

In Cartesian coordinates:

.......................................(1.22)

In Cylindrical coordinates,

....................................(1.23)

In Spherical polar coordinates, Curl operation exhibits the following properties:

..............(1.24)

..............(1.25) Stoke's theorem : It states that the circulation of a vector field provided i.e, and around a closed path is equal to the integral of

over the surface bounded by this path. It may be noted that this equality holds are continuous on the surface.

..............(1.26)

Proof:Let us consider an area S that is subdivided into large number of cells as shown in the figure 1.8

Fig 1.8 Stokes theorem

Let kthcell has surface area and is bounded path Lk while the total area is bounded by path L. As seen from the figure that if we evaluate the sum of the line integrals around the elementary areas, there is cancellation along every interior path and we are left the line integral along path L. Therefore we can write,

..............(1.27) As 0

. which is the stoke's theorem.

.............(1.28)

Coulomb's Law Coulomb's Law may be stated as follows: "The magnitude of the electrostatic force between two point charges is directly proportional to the magnitudes of each charge and inversely proportional to the square of the distance between the charges." Coulomb's law states that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely

proportional to the square of the separation distance between the two objects. In equation form, Coulomb's law can be stated as

(1.29) where Q1 represents the quantity of charge on object 1 (in Coulombs), Q2 represents the quantity of charge on object 2 (in Coulombs), and d represents the distance of separation between the two objects (in meters). The symbol k is a proportionality constant known as the Coulomb's law constant. The value of this constant is dependent upon the medium that the charged objects are immersed in.

Mathematically,

,where k is the proportionality constant.

In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters.

Force F is in Newtons (N) and

,

is called the permittivity of free space.

(We are assuming the charges are in free space. If the charges are any other dielectric medium, we will use instead where dielectric constant of the medium). is called the relative permittivity or the

Therefore

.......................(1.30)

As shown in the Figure 2.1 let the position vectors of the point charges Q1and Q2 are given by and . Let represent the force on Q1 due to charge Q2.

Fig 1.9: Coulomb's Law

The charges are separated by a distance of as

. We define the unit vectors

and can be defined as

..................................(1.31)

. ..(1.32) Similarly the force on Q1 due to charge Q2 can be calculated and if then we can write When we have a number of point charges, to determine the force on a particular charge due to all other charges, we apply principle of superposition. If we have N number of charges Q1,Q2,.........QN located respectively at the points represented by the position vectors ,...... , the force experienced by a charge Q located at is given by, , represents this force

.................................(1.33) Electric Field The electric field intensity or the electric field strength at a point is defined as the force per unit charge. That is

or,

.......................................(1.34)

The electric field intensity E at a point r (observation point) due a point charge Q located at (source point) is given by:

..........................................(1.35)

For a collection of N point charges Q1 ,Q2 ,.........QN located at obtained as

,

,......

, the electric field intensity at point

is

........................................(1.36) The expression (2.6) can be modified suitably to compute the electric filed due to a continuous distribution of charges. In figure 1.10we consider a continuous volume distribution of charge (t) in the region denoted as the source region. For an elementary charge , i.e. considering this charge as point charge, we can write the field expression as: Fig1.10: Continuous Volume Distribution

.............(2.7) When this expression is integrated over the source region, we get the electric field at the point P due to this distribution of charges. Thus the expression for the electric field at P can be written as:

..........................................(1.37) Similar technique can be adopted when the charge distribution is in the form of a line charge density or a surface charge density.

........................................(1.38)

........................................(1.39)

Electric field strength Electric field strength is a vector quantity; it has both magnitude and direction. The magnitude of the electric field strength is defined in terms of how it is measured. Let's suppose that an electric charge can be denoted by the symbol Q. This electric charge creates an electric field; since Q is the source of the electric field, we will refer to it as the source charge. The strength of the source charge's electric field could be measured by any other charge placed somewhere in its surroundings. The charge that is used to measure the electric field strength is referred to as a test charge since it is used to test the field strength. The test charge has a quantity of charge denoted by the symbol q. When placed within the electric field, the test charge will experience an electric force - either attractive or repulsive. As is usually the case, this force will be denoted by the symbol F. The magnitude of the electric field is simply defined as the force per charge on the test charge.

If the electric field strength is denoted by the symbol E, then the equation can be rewritten in symbolic form as

. The standard metric units on electric field strength arise from its definition. Since electric field is defined as a force per charge, its units would be force units divided by charge units. In this case, the standard metric units are Newton/Coulomb or N/C. Electric Field Lines The magnitude or strength of an electric field in the space surrounding a source charge is related directly to the quantity of charge on the source charge and inversely to the distance from the source charge. The direction of the electric field is always directed in the direction that a positive test charge would be pushed or pulled if placed in the space surrounding the source charge. Since electric field is a vector quantity, it can be represented by a vector arrow. For any given location, the arrows point in the direction of the electric field and their length is proportional to the strength of the electric field at that location. Such vector arrows are shown in the diagram below. Note that the length of the arrows are longer when closer to the source charge and shorter when further from the source charge.

A more useful means of visually representing the vector nature of an electric field is through the use of electric field lines of force. Rather than draw countless vector arrows in the space surrounding a source charge, it is perhaps more useful to draw a pattern of several lines which extend between infinity and the source charge. These pattern of lines, sometimes referred to as electric field lines, point in the direction which a positive test charge would accelerate if placed upon the line. As such, the lines are directed away from positively charged source charges and toward negatively charged source charges. To communicate information about the direction of the field, each line must include an arrowhead which points in the appropriate direction. An electric field line pattern could include an infinite number of lines. Because drawing such large quantities of lines tends to decrease the readability of the patterns, the number of lines are usually limited. The presence of a few lines around a charge is typically sufficient to convey the nature of the electric field in the space surrounding the lines.

Electric Fields and Conductors Electrostatic equilibrium is the condition established by charged conductors in which the excess charge has optimally distanced itself so as to reduce the total amount of repulsive forces. Once a charged conductor has reached the state of electrostatic equilibrium, there is no further motion of charge about the surface.

Electric Fields Inside of Charged Conductors Charged conductors which have reached electrostatic equilibrium share a variety of unusual characteristics. One characteristic of a conductor at electrostatic equilibrium is that the electric field anywhere beneath the surface of a charged conductor is zero. If an electric field did exist beneath the surface of a conductor (and inside of it), then the electric field would exert a force on all electrons that were present there. This net force would begin to accelerate and move these electrons. But objects at electrostatic equilibrium have no further motion of charge about the surface. So if this were to occur, then the original claim that the object was at electrostatic equilibrium would be a false claim. If the electrons within a conductor have assumed an equilibrium state, then the net force upon those electrons is zero. The electric field lines either begin or end upon a charge and in the case of a conductor, the charge exists solely upon its outer surface. The lines extend from this surface outward, not inward. This of course presumes that our conductor does not surround a region of space where there was another charge. To illustrate this characteristic, let's consider the space between and inside of two concentric, conducting cylinders of different radii as shown in the diagram at the right. The outer cylinder is charged positively. The inner cylinder is charged negatively. The electric field about the inner cylinder is directed towards the negatively charged cylinder. Since this cylinder does not surround a region of space where there is another charge, it can be concluded that the excess charge resides solely upon the outer surface of this inner cylinder. The electric field inside the inner cylinder would be zero. When drawing electric field lines, the lines would be drawn from the inner surface of the outer cylinder to the outer surface of the inner cylinder. For the excess charge on the outer cylinder, there is more to consider than merely the repulsive forces between charges on its surface. While the excess charge on the outer cylinder seeks to reduce repulsive forces between its excess charge, it must balance this with the tendency to be attracted to the negative charges on the inner cylinder. Since the outer cylinder surrounds a region which is charged, the characteristic of charge residing on the outer surface of the conductor does not apply. This concept of the electric field being zero inside of a closed conducting surface was first demonstrated by Michael Faraday, a 19th century physicist who promoted the field theory of electricity. Faraday constructed a room within a room, covering the inner room with a metal foil. He sat inside the inner room with an electroscope and charged the surfaces of the outer and inner room using an electrostatic generator. While sparks were seen flying between the walls of the two rooms, there was no detection of an electric field within the inner room. The excess charge on the walls of the inner room resided entirely upon the outer surface of the room. The inner room with the conducting frame which protected Faraday from the static charge is now referred to as a Faraday's cage. The cage serves to shield whomever and whatever is on the inside from the influence of electric fields. Any closed, conducting surface can serve as a Faraday's cage, shielding whatever it surrounds from the

potentially damaging affects of electric fields. This principle of shielding is commonly utilized today as we protect delicate electrical equipment by enclosing them in metal cases. Even delicate computer chips and other components are shipped inside of conducting plastic packaging which shields the chips from potentially damaging affects of electric fields. Electric Fields are Perpendicular to Charged Surfaces A second characteristic of conductors at electrostatic equilibrium is that the electric field upon the surface of the conductor is directed entirely perpendicular to the surface. There cannot be a component of electric field (or electric force) that is parallel to the surface. If the conducting object is spherical, then this means that the perpendicular electric field vector are aligned with the center of the sphere. If the object is irregularly shaped, then the electric field vector at any location is perpendicular to a tangent line drawn to the surface at that location. Understanding why this characteristic is true demands an understanding of vectors, force and motion. The motion of electrons, like any physical object, is governed by Newton's laws. One outcome of Newton's laws was that unbalanced forces cause objects to accelerate in the direction of the unbalanced force and a balance of forces cause objects to remain at equilibrium. This truth provides the foundation for the rationale behind why electric fields must be directed perpendicular to the surface of conducting objects. If there were a component of electric field directed parallel to the surface, then the excess charge on the surface would be forced into accelerated motion by this component. If a charge is set into motion, then the object upon which it is on is not in a state of electrostatic equilibrium. Therefore, the electric field must be entirely perpendicular to the conducting surface for objects which are at electrostatic equilibrium. Certainly a conducting object which has recently acquired an excess charge has a component of electric field (and electric force) parallel to the surface; it is this component which acts upon the newly acquired excess charge to distribute the excess charge over the surface and establish electrostatic equilibrium. But once reached, there is no longer any parallel component of electric field and no longer any motion of excess charge.

Electric Fields and Surface Curvature A third characteristic of conducting objects at electrostatic equilibrium is that the electric fields are strongest at locations along the surface where the object is most curved. The curvature of a surface can range from absolute flatness on one extreme to being curved to a blunt point on the other extreme.

A flat location has no curvature and is characterized by relatively weak electric fields. On the other hand, a blunt point has a high degree of curvature and is characterized by relatively strong electric fields. A sphere is uniformly shaped with the same curvature at every location along its surface. As such, the electric field strength on the surface of a sphere is everywhere the same. To understand the rationale for this third characteristic, we will consider an irregularly shaped object which is negatively charged. Such an object has an excess of electrons. These electrons would distribute themselves in such a manner as to reduce the affect of their repulsive forces. Since electrostatic forces vary inversely with the square of the distance, these electrons would tend to position themselves so as to increase their distance from one another. On a regularly shaped sphere, the ultimate distance between every neighboring electron would be the same. But on an irregularly shaped object, excess electrons would tend to accumulate in greater density along locations of greatest curvature. Consider the diagram at the right. Electrons A and B are located along a flatter section of the surface. Like all well-behaved electrons, they repel each other. The repulsive forces are directed along a line connecting charge to charge, making the repulsive force primarily parallel to the surface. On the other hand, electrons C and D are located along a section of the surface with a sharper curvature. These excess electrons also repel each other with a force directed along a line connecting charge to charge. But now the force is directed at a sharper angle to the surface. The components of these forces parallel to the surface are considerably less. A majority of the repulsive force between electrons C and D is directed perpendicular to the surface. The parallel components of these repulsive forces is what causes excess electrons to move along the surface of the conductor. The electrons will move and distribute themselves until electrostatic equilibrium is reached. Once reached, the resultant of all parallel components on any given excess electron (and on all excess electrons) will add up to zero. All the parallel components of force on each of the electrons must be zero since the net force parallel to the surface of the conductor is always zero (the second characteristic discussed above). For the same separation distance, the parallel component

of force is greatest in the case of electrons A and B. So to acquire this balance of parallel forces, electrons A and B must distance themselves further from each other than electrons C and D. Electrons C and D on the other hand can crowd closer together at their location since that the parallel component of repulsive forces is less. In the end, a relatively large quantity of charge accumulates on the locations of greatest curvature. This larger quantity of charge combined with the fact that their repulsive forces are primarily directed perpendicular to the surface results in a considerably stronger electric field at such locations of increased curvature.

The fact that surfaces which are sharply curved to a blunt edge create strong electric fields is the underlying principle for the use of lightning rods. Electric scalar PotentialIn the previous sections we have seen how the electric field intensity due to a charge or a charge distribution can be found using Coulomb's law or Gauss's law. Since a charge placed in the vicinity of another charge (or in other words in the field of other charge) experiences a force, the movement of the charge represents energy exchange. Electrostatic potential is related to the work done in carrying a charge from one point to the other in the presence of an electric field. Let us suppose that we wish to move a positive test charge from a point P to another point Q as shown in the Fig. 1.11 The force at any point along its path would cause the particle to accelerate and move it out of the region if unconstrained. Since we are dealing with an electrostatic case, a force equal to the negative of that acting on the charge is to be applied while moves from P to Q. The work done by this external agent in moving the charge by a distance is given by: Fig 1.11 Movement of Test Charge in Electric Field

.............................(1.40)

The negative sign accounts for the fact that work is done on the system by the external agent.

.....................................(1.41) The potential difference between two points P and Q , VPQ, is defined as the work done per unit charge, i.e.

...............................(1.42) It may be noted that in moving a charge from the initial point to the final point if the potential difference is positive, there is a gain in potential energy in the movement, external agent performs the work against the field. If the sign of the potential difference is negative, work is done by the field. We will see that the electrostatic system is conservative in that no net energy is exchanged if the test charge is moved about a closed path, i.e. returning to its initial position. Further, the potential difference between two points in an electrostatic field is a point function; it is independent of the path taken. The potential difference is measured in Joules/Coulomb which is referred to as Volts. Let us consider a point charge Q as shown in the Fig. 1.12

Fig 1.12 Electrostatic Potential calculation for a point charge

Further consider the two points A and B as shown in the Fig.1.12. Considering the movement of a unit positive test charge from B to A , we can write an expression for the potential difference as:

..................................(1.43) It is customary to choose the potential to be zero at infinity. Thus potential at any point ( rA = r) due to a point charge Q can be written as the amount of work done in bringing a unit positive charge from infinity to that point (i.e. rB = 0).

..................................(1.44) Or, in other words,

..................................(1.45) Let us now consider a situation where the point charge Q is not located at the origin as shown in Fig. 1.13.

Fig 1.13: Electrostatic Potential due a Displaced Charge The potential at a point P becomes

..................................(1.46) So far we have considered the potential due to point charges only. As any other type of charge distribution can be considered to be consisting of point charges, the same basic ideas now can be extended to other types of charge distribution also.

Let us first consider N point charges Q1, Q2,.....QN located at points with position vectors ,...... . The potential at a point having position vector can be written as:

,

..................................(1.47)

or,

...........................................................(1.48)

For continuous charge distribution, we replace point charges Qn by corresponding charge elements or or depending on whether the charge distribution is linear, surface or a volume charge distribution and the summation is replaced by an integral. With these modifications we can write:

For line charge,

..................................(1.49)

For surface charge,

.................................(1.50)

For volume charge,

.................................(1.51)

It may be noted here that the primed coordinates represent the source coordinates and the unprimed coordinates represent field point. Further, in our discussion so far we have used the reference or zero potential at infinity. If any other point is chosen as reference, we can write:

.................................(1.52) where C is a constant. In the same manner when potential is computed from a known electric field we can write:

.................................(1.53) The potential difference is however independent of the choice of reference.

.......................(1.54) We have mentioned that electrostatic field is a conservative field; the work done in moving a charge from one point to the other is independent of the path. Let us consider moving a charge from point P1 to P2 in one path and then from point P2 back to P1 over a different path. If the work done on the two paths were different, a net positive or negative amount of work would have been done when the body returns to its original position P1. In a conservative field there is no mechanism for dissipating energy corresponding to any positive work neither any source is present from which energy could be absorbed in the case of negative work. Hence the question of different works in two paths is untenable, the work must have to be independent of path and depends on the initial and final positions. Since the potential difference is independent of the paths taken, VAB = - VBA , and over a closed path,

.................................(1.55) Applying Stokes's theorem, we can write:

............................(1.56) from which it follows that for electrostatic field, ........................................(1.57) Any vector field that satisfies is called an irrotational field.

From our definition of potential, we can write

.................................(1.58) from which we obtain, ..........................................(1.59)

From the foregoing discussions we observe that the electric field strength at any point is the negative of the poten

gradient at any point, negative sign shows that is directed from higher to lower values of . This gives us ano method of computing the electric field, i. e. if we know the potential function, the electric field may be computed. W may note here that that one scalar function contain all the information that three components of are interrelated by the relation . possible because of the fact that three components of Example: Electric Dipole An electric dipole consists of two point charges of equal magnitude but of opposite sign and separated by a small distance. As shown in figure 1.14, the dipole is formed by the two point charges Q and -Q separated by a distance d , the charges being placed symmetrically about the origin. Let us consider a point P at a distance r, where we are interested to find the field. The potential at P due to the dipole can be written as:

carry, the sa

Fig 1.14 : Electric Dipole

..........................(1.60)

When r1 and r2>>d, we can write Therefore,

and

.

....................................................(1.62) We can write, ...............................................(1.63) The quantity is called the dipole moment of the electric dipole.

Hence the expression for the electric potential can now be written as:

................................(1.64)

It may be noted that while potential of an isolated charge varies with distance as 1/ r that of an electric dipole varies as 1/r2 with distance. If the dipole is not centered at the origin, but the dipole center lies at the potential can be written as: , the expression for

........................(1.65) The electric field for the dipole centered at the origin can be computed as

........................(1.66) is the magnitude of the dipole moment. Once again we note that the electric field of electric dipole varies as 1/r3 where as that of a point charge varies as 1/r2. Electric flux density: As stated earlier electric field intensity or simply Electric field' gives the strength of the field at a particular point. The electric field depends on the material media in which the field is being considered. The flux density vector is defined to be independent of the material media (as we'll see that it relates to the charge that is producing it).For a linear

isotropic medium under consideration; the flux density vector is defined as: ................................................(1.67) We define the electric flux as

.....................................(1.68) Gauss's Law: Gauss's law is one of the fundamental laws of electromagnetism and it states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface.

Fig 1.15 Gauss's Law Let us consider a point charge Q located in an isotropic homogeneous medium of dielectric constant . The flux density at a distance r on a surface enclosing the charge is given by

...............................................(1.69) If we consider an elementary area ds, the amount of flux passing through the elementary area is given by

.....................................(1.70)

But

, is the elementary solid angle subtended by the area

at the location of

Q. Therefore we can write

For a closed surface enclosing the charge, we can write which can seen to be same as what we have stated in the definition of Gauss's Law. Application of Gauss's Law Gauss's law is particularly useful in computing or where the charge distribution has some symmetry. We shall illustrate the application of Gauss's Law with some examples. 1.An infinite line charge As the first example of illustration of use of Gauss's law, let consider the problem of determination of the electric field produced by an infinite line charge of density LC/m. Let us consider a line charge positioned along the z-axis as shown in Fig.1.16(a) (next slide). Since the line charge is assumed to be infinitely long, the electric field will be of the form as shown in Fig. 2.4(b) (next slide).

If we consider a close cylindrical surface as shown in Fig.1.16(a), using Gauss's theorm we can write,

.....................................(1.71) Considering the fact that the unit normal vector to areas S1 and S3 are perpendicular to the electric field, the surface integrals for the top and bottom surfaces evaluates to zero. Hence we can write,

Fig 1.16 Infinite Line Charge

.....................................(1.72)

2. Infinite Sheet of Charge As a second example of application of Gauss's theorem, we consider an infinite charged sheet covering the x-z plane as shown in figure 1.17 Assuming a surface charge density of for the infinite surface charge, if we consider a cylindrical volume having sides can write: placed symmetrically , we

Fig1.17: Infinite Sheet of Charge It may be noted that the electric field strength is independent of distance. This is true for the infinite plane of charge; electric lines of force on either side of the charge will be perpendicular to the sheet and extend to infinity as parallel lines. As number of lines of force per unit area gives the strength of the field, the field becomes independent of distance. For a finite charge sheet, the field will be a function of distance.

..............(1.73)

3. Uniformly Charged Sphere Let us consider a sphere of radius r0 having a uniform volume charge density of v C/m3. To determine everywhere, inside and outside the sphere, we construct Gaussian surfaces of radius r < r0 and r > r0 as shown in Fig. 1.18(a) and Fig.1.18(b). For the region be ; the total enclosed charge will

Fig 1.18 Uniformly Charged Sphere .........................(1.74) By applying Gauss's theorem,

...............(1.75)

Therefore

...............................................(1.76) For the region ; the total enclosed charge will be

....................................................................(1.77) By applying Gauss's theorem,

.....................................................(1.78)

Unit II Static Magnetic FieldIntroduction :

In previous chapters we have seen that an electrostatic field is produced by static or stationary charges. The relationship of the steady magnetic field to its sources is much more complicated. The source of steady magnetic field may be a permanent magnet, a direct current or an electric field changing with time. In this chapter we shall mainly consider the magnetic field produced by a direct current. The magnetic field produced due to time varying electric field will be discussed later. Historically, the link between the electric and magnetic field was established Oersted in 1820. Ampere and others extended the investigation of magnetic effect of electricity . There are two major laws governing the magnetostatic fields are:

Biot-Savart Law Ampere's Law

Usually, the magnetic field intensity is represented by the vector . It is customary to represent the direction of the magnetic field intensity (or current) by a small circle with a dot or cross sign depending on whether the field (or current) is out of or into the page as shown in Fig. 2.1.

(or l ) out of the page

(or l ) into the page

Fig. 2.1: Representation of magnetic field (or current) Biot- Savart Law This law relates the magnetic field intensity dH produced at a point due to a differential current element as shown in Fig. 2.2.

Fig. 2.2: Magnetic field intensity due to a current element

The magnetic field intensity

at P can be written as,

............................(2.1a)

..............................................(2.1b)

where

is the distance of the current element from the point P.

Similar to different charge distributions, we can have different current distribution such as line current, surface current and volume current. These different types of current densities are shown in Fig. 2.3.

Line Current

Surface Current

Volume Current

Fig. 2.3: Different types of current distributions

By denoting the surface current density as K (in amp/m) and volume current density as J (in amp/m2) we can write: ......................................(2.2) ( It may be noted that )

Employing Biot-Savart Law, we can now express the magnetic field intensity H. In terms of these current distributions.

.............................

for

line

current............................(2.3a)

........................ for surface current ....................(2.3b)

....................... for volume current......................(2.3c) To illustrate the application of Biot - Savart's Law, we consider the following example. Example 2.1: We consider a finite length of a conductor carrying a current placed along zaxis as shown in the Fig 2.4. We determine the magnetic field at point P due to this current carrying conductor.

Fig. 2.4: Field at a point P due to a finite length current carrying conductor With reference to Fig. 2.4, we find that ..........................................(2.4) Applying Biot - Savart's law for the current element

we can write,

..............................................(2.5) Substituting we can write,

..............(2.6) We find that, for an infinitely long conductor carrying a current I , Therefore, and

.........................................................................................(2.7)

The value of the constant of proportionality ' K' depends upon a property called permeability of the medium around the conductor. Permeability is represented by symbol 'm' and the constant 'K' is expressed in terms of 'm' as

Magnetic field 'B' is a vector and unless we give the direction of ' dB', its description is not complete. Its direction is found to be perpendicular to the plane of 'r' and 'dl'. If we assign the direction of the current 'I' to the length element 'dl', the vector product dl x r has magnitude r dl sinq and direction perpendicular to 'r' and 'dl'. Hence, BiotSavart law can be stated in vector form to give both the magnitude as well as direction of magnetic field due to a current element as

Value of permeability changes from medium to medium. For ferromagnetic materials it is much higher than that for other materials. The permeability of free space (vacuum) is denoted by the symbol 'm0' and its value is 4p x 107 Wb/Am

Ampere's Circuital Law: Ampere's circuital law states that the line integral of the magnetic field (circulation of H ) around a closed path is the net current enclosed by this path. Mathematically, ......................................(24.8) The total current I enc can be written as,

......................................(24.9) By applying Stoke's theorem, we can write

......................................(2.10) which is the Ampere's law in the point form. Applications of Ampere's law: We illustrate the application of Ampere's Law with some examples. Example2.2: We compute magnetic field due to an infinitely long thin current carrying conductor as shown in Fig. 2.5. Using Ampere's Law, we consider the close path to be a circle of radius as shown in the Fig. 4.5. If we consider a small current element containing both ,i.e., and . , is perpendicular to the plane that will be present is

. Therefore only component of

By applying Ampere's law we can write,

...................................... (2.11)

Therefore,

which is same as equation (2.7)

Fig. 2.5: Magnetic field due to an infinite thin current carrying conductor Example 2.3: We consider the cross section of an infinitely long coaxial conductor, the inner conductor carrying a current I and outer conductor carrying current - I as shown in figure 2.6. We compute the magnetic field as a function of as follows: In the region

......................................(2.12)

............................(2.13) In the region

......................................(2.14)

Fig. 2.6: Coaxial conductor carrying equal and opposite currents

In the region

......................................(2.15)

........................................(2.16) In the region ......................................(2.17)

Lorentz force A charged particle at rest will not interact with a static magnetic field. But if the charged particle is moving in a magnetic field, the magnetic character of a charge in motion becomes evident. It experiences a deflecting force. The force is greatest when the particle moves in a direction perpendicular to the magnetic field lines. At other angles, the force is less and becomes zero when the particles move parallel to the field lines. In any case, the direction of the force is always perpendicular to the magnetic field lines and to the velocity of the charged particle. Magnetic Flux Density The amount of magnetic flux through a unit area taken perpendicular to the direction of the magnetic flux. Also called magnetic induction.

Definition Of Ampere When two current carrying conductors are placed next to each other, we notice that each induces a force on the other. Each conductor produces a magnetic field around itself (BiotSavart law) and the second experiences a force that is given by the Lorentz force.

Ampere's Law The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source. Ampere's law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

Or

In the electric case, the relation of field to source is quantified in Gauss's Law

which is a very powerful tool for calculating electric fields. Application of Ampere's law: Ampere's law can be used to calculate 'B' for various current carrying conductor configurations.

Gauss's Law Gauss's law for magnetic field This law deals with magnetic flux inside a closed surface and is equivalent to Gauss's law for electric field discussed in Electric Charge and Electric Field, connected electric flux j E and electric charge.

And j E = E. A Similarly, magnetic flux fB can be defined as the number of lines of force crossing a unit area. Magnetic flux fB = B.A Since there are no free magnetic charges, the magnetic flux crossing a closed surface will always be zero. Thus Gauss's law of magnetic field says that the net magnetic flux fB out of any closed surface is zero. or B.A = 0 Lenz's law Soon after Faraday proposed his law of electromagnetic induction, Lenz gave the law determining the direction of the induced emf. Lenz's law may be stated as follows: The direction of the induced current is such as to oppose the cause producing it. Lenz's law can be compared with the Newton's third law ? every action has equal and opposite reaction. When an emf is generated by a change in magnetic flux according to the Faraday's law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change that produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the 'B' field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant.

Magnetic Flux Faraday understood that the magnitude of the induced current in a loop was due to the "amount of magnetic field" passing through the loop. To visualize this "amount of magnetic field", which is now called the magnetic flux, he introduced a mental picture of magnetic field as lines of force. This is exactly analogous to electric flux. Magnetic flux is the product of the 'B' times the perpendicular area that it penetrates.

The contribution to jB for a given area is equal to the area times the component of magnetic field perpendicular to the area. For a closed surface, the sum of magnetic flux is always equal to zero (This is also known as Gauss's law for magnetic field). The standard unit for magnetic flux is a weber (Wb), it is the number of magnetic lines of force (Tesla) crossing a unit area (m2).

Magnetic Flux Density: In simple matter, the magnetic flux density where related to the magnetic field intensity as

called the permeability. In particular when we consider the free space

where H/m is the permeability of the free space. Magnetic flux density is measured in terms of Wb/m 2 . The magnetic flux density through a surface is given by:

Wb

......................................(2.18)

In the case of electrostatic field, we have seen that if the surface is a closed surface, the net flux passing through the surface is equal to the charge enclosed by the surface. In case of magnetic field isolated magnetic charge (i. e. pole) does not exist. Magnetic poles always occur in pair (as N-S). For example, if we desire to have an isolated magnetic pole by dividing the magnetic bar successively into two, we end up with pieces each having north (N) and south (S) pole as shown in Fig. 2.7 (a). This process could be continued until the magnets are of atomic dimensions; still we will have N-S pair occurring together. This means that the magnetic poles cannot be isolated.

Fig. 2.7: (a) Subdivision of a magnet (b) Magnetic field/ flux lines of a straight current carrying conductor Similarly if we consider the field/flux lines of a current carrying conductor as shown in Fig. 2.7 (b), we find that these lines are closed lines, that is, if we consider a closed surface, the number of flux lines that would leave the surface would be same as the number of flux lines that would enter the surface. From our discussions above, it is evident that for magnetic field,

......................................(2.19) which is the Gauss's law for the magnetic field. By applying divergence theorem, we can write:

Hence,

......................................(2.20)

which is the Gauss's law for the magnetic field in point form. Magnetic Scalar and Vector Potentials: In studying electric field problems, we introduced the concept of electric potential that simplified the computation of electric fields for certain types of problems. In the same manner let us relate the magnetic field intensity to a scalar magnetic potential and write:

...................................(2.21) From Ampere's law , we know that

......................................(2.22) Therefore, But using vector identity, we find that ............................(2.23) is valid only where . Thus the scalar magnetic

potential is defined only in the region where

. Moreover, Vm in general is not a single valued function of position.

This point can be illustrated as follows. Let us consider the cross section of a coaxial line as shown in fig 2.8. In the region , and

Fig. 2.8: Cross Section of a Coaxial Line

If Vm is the magnetic potential then,

If we set Vm = 0 at

then c=0 and

We observe that as we make a complete lap around the current carrying conductor , we reach again but Vm this time becomes

We observe that value of Vm keeps changing as we complete additional laps to pass through the same point. We introduced Vm analogous to electostatic potential V. But for static electric fields, but and even if , whereas for steady magnetic field along the path of integration. wherever

We now introduce the vector magnetic potential which can be used in regions where current density may be zero or nonzero and the same can be easily extended to time varying cases. The use of vector magnetic potential provides elegant ways of solving EM field problems.

Since write

and we have the vector identity that for any vector .

,

, we can

Here, the vector field can find

is called the vector magnetic potential. Its SI unit is Wb/m. Thus if can be found from and related its curl to through a curl operation. . A vector function is is made as follows.

of a given current distribution,

We have introduced the vector function

defined fully in terms of its curl as well as divergence. The choice of

...........................................(2.24)

By using vector identity,

.................................................(2.25) .........................................(2.26)

Great deal of simplification can be achieved if we choose

.

Putting , we get which is vector poisson equation. In Cartesian coordinates, the above equation can be written in terms of the components as ......................................(2.27a) ......................................(2.27b) ......................................(2.27c) The form of all the above equation is same as that of

..........................................(2.28) for which the solution is

..................(2.29)

In case of time varying fields we shall see that , which is known as Lorentz condition, V being the electric potential. Here we are dealing with static magnetic field, so . By comparison, we can write the solution for Ax as

...................................(2.30) Computing similar solutions for other two components of the vector potential, the vector potential can be written as

.......................................(2.31)

This equation enables us to find the vector potential at a given point because of a volume current density . Similarly for line or surface current density we can write

...................................................(2.32)

respectively. ..............................(2.33) The magnetic flux through a given area S is given by

.............................................(2.34) Substituting

.........................................(2.35) Vector potential thus have the physical significance that its integral around any closed path is equal to the magnetic flux passing through that path.

Magnetic Moment In A Magnetic Field The magnetic moment of an object is a vector relating the aligning torque in a magnetic field experienced by the object to the field vector itself. The relationship is given by

where is the torque, measured in newton-meters, is the magnetic moment, measured in ampere meters-squared, and is the magnetic field, measured in teslas or, equivalently in newtons per (ampere-meter). Magnetic Scalar Potential The magnetic scalar potential is another useful tool in describing the magnetic field around a current source. It is only defined in regions of space in absence of (but could be near) currents. The magnetic scalar potential is defined by the equation:

Applying Ampere's Law to the above definition we get:

Since in any continuous field, the curl of a gradient is zero, this would suggest that magnetic scalar potential fields cannot support any sources. In fact, sources can be supported by applying discontinuities to the potential field (thus the same point can have two values for points along the disconuity). These discontinuities are also known as "cuts". When solving magnetostatics problems using magnetic scalar potential, the source currents must be applied at the discontinuity. The magnetic scalar potential is suited to use around lines/loops of currents, but not a region of space with finite current density. The use of magnetic potential reduces the three components of the magnetic field to one component , making computations and algebraic manipulations easier. It is often used in magnetostatics, but rarely used in other applications. Magnetic Vector Potential The magnetic vector potential is a three-dimensional vector field whose curl is the magnetic field in the theory of electromagnetism:

Since the magnetic field is divergence free (i.e.

),

always exists.

Unit IIIPoisson's equation

Electric And Magnetic Fiels In Materials

The derivation of Poisson's equation in electrostatics follows. SI units are used and Euclidean space is assumed. Starting with Gauss' law for electricity (also part of Maxwell's equations) in a differential control volume, we have:

means to take the divergence. is the electric displacement field. is the charge density. Assuming the medium is linear, isotropic, and homogeneous then: is the permittivity of the the medium. is the electric field. is the vacuum permittivity. is the relative permittivity of the medium. By substitution and division, we have:

Since the curl of the electric field is zero, it is defined by a scalar electric potential field,

Eliminating

by substitution, we have a form of the Poisson equation:

Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then Laplace's equation results.

Laplace's equation In three dimensions, the problem is to find twice-differentiable real-valued functions, of real variables, x, y, and z, such that

This is often written as

or

where div is the divergence, and grad is the gradient, or

where is the Laplace operator. Solutions of Laplace's equation are called harmonic functions. If the right-hand side is specified as a given function, f(x, y, z), i.e.

then the equation is called "Poisson's equation." Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The partial differential operator, , or , (which may be defined in any number of dimensions) is called the Laplace operator, or just the LaplacianFor electrostatic field, we have seen that

..........................................................................................(3.1) Form the above two equations we can write ..................................................................(3.2) Using vector identity we can write, ................(3.3)

For a simple homogeneous medium,

is constant and

. Therefore,

................(3.4) This equation is known as Poissons equation. Here we have introduced a new operator, ( del square), called the Laplacian operator. In Cartesian coordinates,

...............(3.4) Therefore, in Cartesian coordinates, Poisson equation can be written as:

...............(3.5) In cylindrical coordinates,

...............(3.6) In spherical polar coordinate system,

...............(3.7) At points in simple media, where no free charge is present, Poissons equation reduces to ...................................(3.8) which is known as Laplaces equation. Laplaces and Poissons equation are very useful for solving many practical electrostatic field problems where only the electrostatic conditions (potential and charge) at some boundaries are known and solution of electric field and potential is to be found throughout the volume. We shall consider such applications in the section where we deal with boundary value problems.

Polarization density in Maxwell's equations The behavior of electric fields (E, D),magnetic fields (B, H), charge density () and current density (J) are described by Maxwell's equations. The role of the polarization density P is described below.

Relations between E, D and P The polarization density P defines the electric displacement field D as which is convenient for various calculations.A relation between P and E exists in many materials, as described later in the article. Bound charge Electric polarization corresponds to a rearrangement of the bound electrons in the material, which creates an additional charge density, known as the bound charge density b:so that the total charge density that enters Maxwell's equations is given bywhere f is the free charge density (describing charges brought from outside).At the surface of the polarized material, the bound charge appears as a surface charge densitywhere is the normal vector. If P is uniform inside the material, this surface charge is the only bound charge. When the polarization density changes with time, the time-dependent bound-charge density creates a current density of so that the total current density that enters Maxwell's equations is given by where Jf is the free-charge current density, and the second term is a contribution from the magnetization (when it exists).Capacitance and Capacitors

Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. The most common form of charge storage device is a two-plate capacitor. If the charges on the plates are +Q and -Q, and V gives the voltage difference between the plates, then the capacitance is given by

The SI unit of capacitance is the farad; 1 farad = 1 coulomb per volt. Capacitors The capacitance of the majority of capacitors used in electronic circuits is several orders of magnitude smaller than the farad. The most common subunits of capacitance in use today are the millifarad (mF), microfarad (F), the nanofarad (nF) and the picofarad (pF) The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. For example, the capacitance of a parallel-plate capacitor constructed of two parallel plates of area A separated by a distance d is approximately equal to the following:

or

where C is the capacitance in farads, F s is the static permittivity of the insulator used (or 0 for a vacuum) A is the area of each plate, measured in square metres r is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates, (vacuum =1) d is the separation between the plates, measured in metres The equation is a good approximation if d is small compared to the other dimensions of the plates.We have already stated that a conductor in an electrostatic field is an Equipotential body and any charge given to such conductor will distribute themselves in such a manner that electric field inside the conductor vanishes. If an additional amount of charge is supplied to an isolated conductor at a given potential, this additional charge will increase the surface charge density . Since the potential of the conductor is given by , the same. Thus we can write

potential of the conductor will also increase maintaining the ratio

where the constant of proportionality C is called the capacitance of the isolated conductor. SI unit of capacitance is Coulomb/ Volt also called Farad denoted by F. It can It can be seen that if V=1, C = Q. Thus capacity of an isolated conductor can also be defined as the amount of charge in Coulomb required to raise the potential of the conductor by 1 Volt. Of considerable interest in practice is a capacitor that consists of two (or more) conductors carrying equal and opposite charges and separated by some dielectric media or free space. The conductors may have arbitrary shapes. A two-conductor capacitor is shown in figure 3.1

Fig 3.1: Capacitance and Capacitors When a d-c voltage source is connected between the conductors, a charge transfer occurs which results into a positive charge on one conductor and negative charge on the other conductor. The conductors are equipotential surfaces and the field lines are perpendicular to the conductor surface. If V is the mean potential difference between the conductors, the capacitance is given by . Capacitance of a capacitor depends on the geometry of the conductor and the permittivity of the medium between them and does not depend on the charge or potential difference between conductors. The capacitance can be computed by assuming Q(at the same time -Q on the other conductor), first determining using Gausss theorem and then determining example of a parallel plate capacitor Example: Parallel plate capacitor . We illustrate this procedure by taking the

Fig 3.2: Parallel Plate Capacitor For the parallel plate capacitor shown in the figure 3.2, let each plate has area A and a distance h separates the plates. A dielectric of permittivity fills the region between the plates. The electric field lines are confined between the plates. We ignore the flux fringing at the edges of the plates and charges are assumed to be uniformly distributed over the conducting plates with densities By Gausss theorem we can write, and , . .......................(3.9)

As we have assumed

to be uniform and fringing of field is neglected, we see that E is . Thus,

constant in the region between the plates and therefore, we can write for a parallel plate capacitor we have, ........................(3.10) Series and parallel Connection of capacitors

Capacitors are connected in various manners in electrical circuits; series and parallel connections are the two basic ways of connecting capacitors. We compute the equivalent capacitance for such connections. Series Case: Series connection of two capacitors is shown in the figure 3.3. For this case we can write,

.......................(3.11)

Fig 3.3: Series Connection of Capacitors

Fig 3.4: Parallel Connection of Capacitors

The same approach may be extended to more than two capacitors connected in series. Parallel Case: For the parallel case, the voltages across the capacitors are the same. The total charge

Therefore,

.......................(3.12)

Electrostatic Energy and Energy Density We have stated that the electric potential at a point in an electric field is the amount of work required to bring a unit positive charge from infinity (reference of zero potential) to that point. To determine the energy that is present in an assembly of charges, let us first determine the amount of work required to assemble them. Let us consider a number of discrete charges Q1, Q2,......., QN are brought from infinity to their present position one by one. Since initially there is no field present, the amount of work done in bring Q1 is zero. Q2 is brought in the presence of the field of Q1, the work done W1= Q2V21 where V21 is the potential at the location of Q2 due to Q1. Proceeding in this manner, we can write, the total work done ..................................... ............(3.13) Had the charges been brought in the reverse order,

.................(3.14) Therefore,

.... ............(3.15) Here VIJ represent voltage at the Ith charge location due to Jth charge. Therefore,

Or,

................(3.16)

If instead of discrete charges, we now have a distribution of charges over a volume v then we can write,

................(3.17) where Since, is the volume charge density and V represents the potential function. , we can write

.......................................(3.18) Using the vector identity, , we can write

................(3.19)

In the expression the term V as varies as

, for point charges, since V varies as

and D varies as

,

while the area varies as r2. Hence the integral term varies at least ) the integral term tends to zero

and the as surface becomes large (i.e.

Thus the equation for W reduces to

................(3.20)

, is called the energy density in the electrostatic field. Boundary conditions for Electrostatic fields In our discussions so far we have considered the existence of electric field in the homogeneous medium. Practical electromagnetic problems often involve media with different physical properties. Determination of electric field for such problems requires the knowledge of the relations of field quantities at an interface between two media. The conditions that the fields must satisfy at the interface of two different media are referred to as boundary conditions . In order to discuss the boundary conditions, we first consider the field behavior in some common material media. In general, based on the electric properties, materials can be classified into three categories: conductors, semiconductors and insulators (dielectrics). In conductor , electrons in the outermost shells of the atoms are very loosely held and they migrate easily from one atom to the other. Most metals belong to this group. The electrons in the atoms of insulators or dielectrics remain confined to their orbits and under normal circumstances they are not liberated under the influence of an externally applied field. The electrical properties of

semiconductors fall between those of conductors and insulators since semiconductors have very few numbers of free charges. The parameter conductivity is used characterizes the macroscopic electrical property of a material medium. The notion of conductivity is more important in dealing with the current flow and hence the same will be considered in detail later on. If some free charge is introduced inside a conductor, the charges will experience a force due to mutual repulsion and owing to the fact that they are free to move, the charges will appear on the surface. The charges will redistribute themselves in such a manner that the field within the conductor is zero. Therefore, under steady condition, inside a conductor From Gauss's theorem it follows that = 0 .......................(3.21) The surface charge distribution on a conductor depends on the shape of the conductor. The charges on the surface of the conductor will not be in equilibrium if there is a tangential component of the electric field is present, which would produce movement of the charges. Hence under static field conditions, tangential component of the electric field on the conductor surface is zero. The electric field on the surface of the conductor is normal everywhere to the surface . Since the tangential component of electric field is zero, the conductor surface is an equipotential surface. As = 0 inside the conductor, the conductor as a whole has the same potential. We may further note that charges require a finite time to redistribute in a conductor. However, this time is very small sec for good conductor like copper. Let us now consider an interface between a conductor and free space as shown in the figure 3.5. .

Fig 3.5: Boundary Conditions for at the surface of a Conductor Let us consider the closed path pqrsp for which we can write,

.................................(3.22) For and noting that inside the conductor is zero, we can write =0.......................................(3.23) Et is the tangential component of the field. Therefore we find that

Et = 0 ...........................................(3.24) In order to determine the normal component En, the normal component of of the conductor, we consider a small cylindrical Gaussian surface . Let area of the top and bottom faces and as zero, , we approach the surface of the conductor. Since , at the surface represent the

represents the height of the cylinder. Once again, = 0 inside the conductor is

.............(3.25)

..................(3.26) Therefore, we can summarize the boundary conditions at the surface of a conductor as: Et = 0 ........................(3.27)

.....................(3.28) Behavior of dielectrics in static electric field: Polarization of dielectric Here we briefly describe the behavior of dielectrics or insulators when placed in static electric field. Ideal dielectrics do not contain free charges. As we know, all material media are composed of atoms where a positively charged nucleus (diameter ~ 10 -15m) is surrounded by negatively charged electrons (electron cloud has radius ~ 10 -10m) moving around the nucleus. Molecules of dielectrics are neutral macroscopically; an externally applied field causes small displacement of the charge particles creating small electric dipoles.These induced dipole moments modify electric fields both inside and outside dielectric material. Molecules of some dielectric materials posses permanent dipole moments even in the absence of an external applied field. Usually such molecules consist of two or more dissimilar atoms and are called polar molecules. A common example of such molecule is water molecule H2O. In polar molecules the atoms do not arrange themselves to make the net dipole moment zero. However, in the absence of an external field, the molecules arrange themselves in a random manner so that net dipole moment over a volume becomes zero. Under the influence of an applied electric field, these dipoles tend to align themselves along the field. There are some materials that can exhibit net permanent dipole moment even in the absence of applied field. These materials are called electrets that made by heating certain waxes or plastics in the presence of electric field. The applied field aligns the polarized molecules when the material is in the heated state and they are frozen to their new position when after the temperature is brought down to its normal temperatures. Permanent polarization remains without an externally applied field.

As a measure of intensity of polarization, polarization vector

(in C/m2) is defined as:

.......................(3.29) n being the number of molecules per unit volume i.e. dielectric material having polarization dv'. is the dipole moment per unit volume. Let us now consider a

and compute the potential at an external point O due to an elementary dipole

With reference to the figure 2.16, we can write: ..........................................(3.30) Therefore,

....................(3.31))

Fig 3.6: Potential at an External Point due to an Elementary Dipole dv'.

........(3.32) where x,y,z represent the coordinates of the external point O and x',y',z' are the coordinates of the source point.

From the expression of R, we can verify that

.............................................(3.33)

.........................................(3.34) Using the vector identity, ,where f is a scalar quantity , we have,

.......................(3.35) Converting the first volume integral of the above expression to surface integral, we can write

.................(3.36) where is the outward normal from the surface element ds' of the dielectric. From the above expression we find that the electric potential of a polarized dielectric may be found from the contribution of volume and surface charge distributions having densities ......................................................................(3.37) ......................(3.38) These are referred to as polarisation or bound charge densities. Therefore we may replace a polarized dielectric by an equivalent polarization surface charge density and a polarization volume charge density. We recall that bound charges are those charges that are not free to move within the dielectric material, such charges are result of displacement that occurs on a molecular scale during polarization. The total bound charge on the surface is

......................(3.39) The charge that remains inside the surface is

......................(3.40) The total charge in the dielectric material is zero as

......................(3.41) If we now consider that the dielectric region containing charge density charge density becomes ....................(3.42) Since we have taken into account the effect of the bound charge density, we can write the total volume

....................(3.43) Using the definition of we have

....................(3.44) Therefore the electric flux density When the dielectric properties of the medium are linear and isotropic, polarisation is directly proportional to the applied field strength and ........................(3.45) is the electric susceptibility of the dielectric. Therefore, .......................(3.46) is called relative permeability or the dielectric constant of the medium. the absolute permittivity. A dielectric medium is said to be linear when is independent of is called

and the medium is

homogeneous if is also independent of space coordinates. A linear homogeneous and isotropic medium is called a simple medium and for such medium the relative permittivity is a constant. Dielectric constant may be a function of space coordinates. For anistropic materials, the dielectric constant is different in different directions of the electric field, D and E are related by a permittivity tensor which may be written as:

.......................(3.47) For crystals, the reference coordinates can be chosen along the principal axes, which make off diagonal elements of the permittivity matrix zero. Therefore, we have

.......................(3.48) Media exhibiting such characteristics are called biaxial. Further, if called uniaxial. It may be noted that for isotropic media, . then the medium is

Lossy dielectric materials are represented by a complex dielectric constant, the imaginary part of which provides the power loss in the medium and this is in general dependant on frequency. Another phenomenon is of importance is dielectric breakdown. We observed that the applied electric field causes small displacement of bound charges in a dielectric material that results into polarization. Strong field can pull electrons completely out of the molecules. These electrons being accelerated under influence of electric field will collide with molecular lattice structure causing damage or distortion of material. For very strong fields, avalanche breakdown may also occur. The dielectric under such condition will become conducting. The maximum electric field intensity a dielectric can withstand without breakdown is referred to as the dielectric strength of the material. Boundary Conditions for Electrostatic Fields: Let us consider the relationship among the field components that exist at the interface between two dielectrics as shown in the figure 3.7. The permittivity of the medium 1 and medium 2 are Coulomb/m. and respectively and the interface may also have a net charge density

Fig 3.7: Boundary Conditions at the interface between two dielectrics We can express the electric field in terms of the tangential and normal components

..........(3.49) where Et and En are the tangential and normal components of the electric field respectively. Let us assume that the closed path is very small so that over the elemental path length the variation of E can be neglected. Moreover very near to the interface, . Therefore

.......................(3.50)

Thus, we have,

or i.e. the tangential component of an electric field is continuous across the interface. For relating the flux density vectors on two sides of the interface we apply Gausss law to a small pillbox volume as shown in the figure. Once again as , we can write

..................(3.51a) i.e., i.e., .................................................(3.51b) .......................(3.51c)

Thus we find that the normal component of the flux density vector D is discontinuous across an interface by an amount of discontinuity equal to the surface charge density at the interface. Example Two further illustrate these points; let us consider an example, which involves the refraction of D or E at a charge free dielectric interface as shown in the figure 3.8. Using the relationships we have just derived, we can write

.......................(3.52a) .......................(3.52b) In terms of flux density vectors,

.......................(3.53a) .......................(3.53b)

Therefore,

.......................(3.54)

Fig 3.8: Refraction of D or E at a Charge Free Dielectric Interface

Energy The energy (measured in joules) stored in a capacitor is equal to the work done to charge it. Consider a capacitance C, holding a charge +q on one plate and -q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW:

where W is the work measured in joules q is the charge measured in coulombs C is the capacitance, measured in farads We can find the energy stored in a capacitance by integrating this equation. Starting with an uncharged capacitance (q=0) and moving charge from one plate to the other until the plates have charge +Q and -Q requires the work W:

Combining this with the above equation for the capacitance of a flat-plate capacitor, we get:

. where

W is the energy measured in joules C is the capacitance, measured in farads V is the voltage measured in volts Capacitance and 'displacement current'

The physicist James Clerk Maxwell invented the concept of displacement current, , to make Ampre's law consistent with conservation of charge in cases where charge is accumulating, for example in a capacitor. He interpreted this as a real motion of charges, even in vacuum, where he supposed that it corresponded to motion of dipole charges in the ether. Although this interpretation has been abandoned, Maxwell's correction to Ampre's law remains valid (a changing electric field produces a magnetic field). Maxwell's equation combining Ampre's law with the displacement current concept is given as . (Integrating both sides, the integral of can be

replaced courtesy of Stokes's theorem with the integral of over a closed contour, thus demonstrating the interconnection with Ampre's formulation Electric Field Boundary Conditions

Steps to solve boundary condition problems: Typically you are given or have previously calculated the electric field (E) or flux density (D) in one of the two regions. 1) Break the electric flux density vector (D) into tangential and normal components as shown above. 2) Solve for the tangential components like this:D1t D2t = 1 2 or E1t = E2t

Because D = E = o r E

3) Solve for the normal components like this: The normal components depend on the surface charge density s (C/m2) . D1n D2n = s (C/m2) OR 1E1n - 2E2n = s

Special Cases: Perfect Dielectrics (conductivity = 0) Surface charge density can only exist on a conductive surface, so if both materials are perfect dielectrics (have no conductivity), then s = 0. Perfect Conductors (conductivity is infinite) (metals) The electric field inside the metal = 0, so Et = 0 inside the metal, and on its surface. TANGENTIAL E = 0 on surface of metal Magnetic Field Boundary Conditions Use the same figure as above, but replace electric fields or flux density with magnetic fields (H) or flux density (B). Steps to solve boundary condition problems: Typically you are given or have previously calculated the magnetic field (H) or flux density (B) in one of the two regions. 1) Break the magnetic flux density vector (B) into tangential and normal components as shown above. 2) Solve for the tangential components like this: The tangential magnetic fields depend on the surface current density (most books call this Js, some call it K). This is the current density (A/m 2 ) flowing ON THE SURFACE.H 2t H 1t = J s ( A / m 2 ) = K B = H

3) Solve for the normal components like this: B1n = B2n OR 1H1n = 2 H2n Special Cases: Perfect Dielectrics (conductivity = 0) Surface current density can only exist on a conductive surface, so if both materials are perfect dielectrics (have no conductivity), then Js = 0 Perfect Conductors (conductivity is infinite) (metals) The magnetic field inside the metal = 0, so Hn = 0 inside the metal, and on its surface. NORMAL H = 0 on surface of metal. Continuity Equation

The continuity equation is derived from two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density,

Derivation One of Maxwell's equations, Ampre's law, states that

Taking the divergence of both sides results in

but the divergence of a curl is zero, so that

Another one of Maxwell's equations, Gauss's law, states that

Substitute this into equation (1) to obtain

which is the continuity equation.

Current Density and Ohm's Law:

In our earlier discussion we have mentioned that, conductors have free electrons that move randomly under thermal agitation. In the absence of an external electric field, the average thermal velocity on a microscopic scale is zero and so is the net current in the conductor. Under the influence of an applied field, additional velocity is superimposed on the random velocities. While the external field accelerates the electron in a direction opposite to it, the collision with atomic lattice however provide the frictional mechanism by which the electrons lose some of the momentum gained between the collisions. As a result, the electrons move with some average drift velocity field by the relationship . This drift velocity can be related to the applied electric

......................(3.55) where is the average time between the collisions.

The quantity

i.e., the the drift velocity per unit applied field is called the mobility of .

electrons and denoted by

Thus , e is the magnitude of the electronic charge and drifts opposite to the applied field.

, as the electron

Let us consider a conductor under the influence of an external electric field. If the number of electrons per unit volume, then the charge normal to the direction of the drift velocity is given by: crossing an area

represents that is

........................................(3.56) This flow of charge constitutes a current across , which is given by,

................(3.57) The conduction current density can therefore be expressed as

.................................(3.58) where is called the conductivity. In vector form, we can write, ..........................................................(3.59)

The above equation is the alternate way of expressing Ohm's law and this relations