Mathematics for Electromagnetism (PHY205)

 

The following provides a brief description and summary of the mathematics required for PHY205, Electromagnetism. Its main objective is to provide a revision of the most important and relevant points. It is not intended to teach the mathematics from scratch. For more detailed treatments consult your first year mathematics’ notes or textbooks. Some of the electromagnetism textbooks provide a chapter or appendix covering the required mathematics.

 

Partial Differentiation

Many physical quantities are a function of more than one variable (e.g. the pressure of a gas depends upon both temperature and volume, a magnetic field may be a function of the three spatial co-ordinates (x,y,z) and time (t)). Hence when differentiating a function there is usually a choice of which variable we differentiate with respect to.

For example consider the function f which depends upon the variables x and y (f(x,y)). We can differentiate f with respect to x or y. When we differentiate with respect to a given variable we proceed in the same manner as in basic differentiation for functions which depend upon one variable. However for functions of more than one variable all other variables are treated as if they are constants. The symbols for the differential are modified (‘¶’ is used instead of ‘d’).

The derivatives of f with respect to x or y are written as 

respectively.

  Example f(x,y)=3x3+yx2+4xy2+5y3

 

  We can also take higher order derivatives, e.g. differentiate twice with respect to x

  or differentiate with respect to one variable (say x ) and then a second (y)

  note that for all well behaved functions the order in which we differentiate (e.g. x then y or y then x) is unimportant, i.e.

Angles and Solid Angles

We need to consider both normal (one-dimensional) and solid (two-dimensional angles)

In (a) the line of length dl has a component dlcosq along the arc of the circle. The angle is defined as this component divided by the radius of the circle r.

The units are radians. As the circumference of a circle is 2pr there are 2pr/r=2p radians in a circle.

 

In (b) the area da makes an angle q to the surface of the sphere radius r. The projection of da onto the surface of the sphere is hence dacosq and the solid angle is defined as this projection divided by the square of the radius

The units are steradians. As the area of a sphere is 4pr2 there are 4pr2/r2=4p steradians in a sphere.

 

VectorsMany physical quantities have a direction as well as a magnitude. Examples are force velocity and, in electromagnetism, electric and magnetic fields. We

describe such quantities using vectors. At each point in space we can imagine an arrow whose length gives the magnitude of the quantity it describes and whose direction corresponds to the direction of the quantity.

 

Vector components

In dealing with vectors it is often convenient to describe a vector in terms of components. Because space has three-dimensions, three components lying along three orthogonal directions are required to describe any vector. The most common system is the Cartesian one where the three directions are the x, y and z-axes.

To define any vector A in the Cartesian system we need the size of the components along the three axes (Ax, Ay and Az) and

three unit vectors that are parallel to the three axes (i, j and k parallel to x, y and z respectively).

  The vector A is given by

  The magnitude of A is given by

 

Non-Cartesian Systems

Although the Cartesian system is the most common one and the easiest to visualise and use, There are

two other system that are useful when considering problems with cylindrical or spherical symmetry.

In the cylindrical system the three components of the vector are defined as lying along the radial direction in the x-y plane (r), the angle between the projection onto the x-y plane and the x-axis (f) and the vertical or z component (z).

  In the spherical system the three components are along the radial direction (r), the angle between the projection onto the x-y plane and the x-axis (f) and the angle between the vector and the z-axis (q).

 

Multiplication of VectorsThere are two types of multiplication, the dot product, resulting in a scalar and the vector product resulting in a vector.

 

The dot product of two vectors

If we have two vectors A and B then the dot product of A and B is defined as

 

  where A and B are the magnitudes of vectors A and B and q is the angle between the two vectors.

  The dot product is commutative

 

  Physically the dot product represents the projection of one of the vectors on to the other times the magnitude of the other. If the two vectors are mutually perpendicular then the dot product is zero (cos90°=0).

  In Cartesian co-ordinates, if we have two vectors A and B with components (Ax, Ay, Az) and (Bx, By, Bz) respectively then

 

 

Physical application of the dot product

We know from mechanics that if a force F moves through a distance L then the work done is equal to the component of the force along the direction of movement multiplied by L. In the diagram below the component of F along the direction of movement is Fcosq. Hence the work done is FLcosq. However if we use vectors F and L to describe the force and the movement respectively we have from the definition of the dot product . Hence when a force F is moved by a distance L the work done is simply the dot product of the two vectors. This is an application of the dot product which we will

use many times in the electromagnetism course.

 

  The cross product of two vectors

If we have two vectors A and B then the cross product of A and B is defined as

  A´B = ABsinqn

  Where n is a unit vector normal to the plane containing the two vectors A and B and whose direction is given by the right-hand rule.

  The cross product of two vectors is not commutative as sin(-q)=-sinq.

A´B = - B´A

  In Cartesian co-ordinates the cross product of the vectors A and B with components (Ax, Ay, Az) and (Bx, By, Bz) is given by

  A´B = (AyBz-AzBy)i+(AzBx-AxBz)j+(AxBy-AyBx)k

 

  The cross product of a vector with itself is zero as the angle between the two vectors is 0 and sin(0)=0.

  Physical application of the cross product

A force F acts at a distance r from a point of rotation. The torque (T)about this point is the distance from where F acts to the point of

rotation (r) multiplied by the normal component of F. T=rFsinq, where q is the angle between F and the line drawn through the point of rotation. However if we define torque in terms of a vector whose magnitude gives the size of the torque and whose direction points along the axis of rotation then

T= r´F where r is the vector from the point of rotation to the point where F acts. The direction of T gives the sense of rotation from the right-hand screw rule.

 

Calculus of scalars and vectorsMuch of physics is concerned with how one quantity varies when one or more other quantities change. As in other areas of physics, in electromagnetism we will be concerned with the spatial variation (or derivative) of both scalar and vector quantities.

  The mathematics can be summarised by the use of a differential operator called 'del' (symbol Ñ) which itself has directional properties.

  In Cartesian co-ordinates Ñ is given by

  There are three physically meaningful ways in which Ñ can be applied to scalars and vectors: it can be applied to a scalar to give a vector (gradient), it can form the dot product with a vector to give a scalar (divergence) and it can form the cross product with a vector to give another vector (curl).

 

Gradient Ñ

The gradient of a scalar function f is written Ñf or grad f and is given in the Cartesian system by

 

  The resultant quantity is a vector.

  e.g. if f(x,y,z)=2x2+y3+z2xy then Ñf=i(4x+z2y)+j(3y2+z2x)+k2xyz

  Physical significance of the gradient. At any point the gradient of a function points in the direction corresponding to that for which the function varies most rapidly. The magnitude of the gradient vector gives the size of this maximum variation.

  Example If f(x,y) gives the height (or alternatively the z co-ordinate) of a surface as a function of the x and y co-ordinates then at any point Ñf will point in the direction of maximum slope of the surface.

A small ball placed on the surface will tend to roll along the direction opposite to Ñf. The gravitational force acting on the ball is -mgÑf where m is its mass and g is the acceleration due to gravity. The negative sign arises because the force acts in the opposite direction to Ñf. Alternatively if we define U(x,y) as the gravitational potential energy of the ball (U(x,y)=mg f(x,y)) then the gravitational force = -ÑU. This is a general result: Force = -gradient(potential energy). The potential energy may be gravitational, electrical etc.

Mathematics for Electromagnetism (PHY205)

 

The following provides a brief description and summary of the mathematics required for PHY205, Electromagnetism. Its main objective is to provide a revision of the most important and relevant points. It is not intended to teach the mathematics from scratch. For more detailed treatments consult your first year mathematics’ notes or textbooks. Some of the electromagnetism textbooks provide a chapter or appendix covering the required mathematics.

 

Partial Differentiation

Many physical quantities are a function of more than one variable (e.g. the pressure of a gas depends upon both temperature and volume, a magnetic field may be a function of the three spatial co-ordinates (x,y,z) and time (t)). Hence when differentiating a function there is usually a choice of which variable we differentiate with respect to.

For example consider the function f which depends upon the variables x and y (f(x,y)). We can differentiate f with respect to x or y. When we differentiate with respect to a given variable we proceed in the same manner as in basic differentiation for functions which depend upon one variable. However for functions of more than one variable all other variables are treated as if they are constants. The symbols for the differential are modified (‘¶’ is used instead of ‘d’).

The derivatives of f with respect to x or y are written as 

respectively.

  Example f(x,y)=3x3+yx2+4xy2+5y3

 

  We can also take higher order derivatives, e.g. differentiate twice with respect to x

  or differentiate with respect to one variable (say x ) and then a second (y)

  note that for all well behaved functions the order in which we differentiate (e.g. x then y or y then x) is unimportant, i.e.

Angles and Solid Angles

We need to consider both normal (one-dimensional) and solid (two-dimensional angles)

In (a) the line of length dl has a component dlcosq along the arc of the circle. The angle is defined as this component divided by the radius of the circle r.

The units are radians. As the circumference of a circle is 2pr there are 2pr/r=2p radians in a circle.

 

In (b) the area da makes an angle q to the surface of the sphere radius r. The projection of da onto the surface of the sphere is hence dacosq and the solid angle is defined as this projection divided by the square of the radius

The units are steradians. As the area of a sphere is 4pr2 there are 4pr2/r2=4p steradians in a sphere.

 

VectorsMany physical quantities have a direction as well as a magnitude. Examples are force velocity and, in electromagnetism, electric and magnetic fields. We

describe such quantities using vectors. At each point in space we can imagine an arrow whose length gives the magnitude of the quantity it describes and whose direction corresponds to the direction of the quantity.

 

Vector components

In dealing with vectors it is often convenient to describe a vector in terms of components. Because space has three-dimensions, three components lying along three orthogonal directions are required to describe any vector. The most common system is the Cartesian one where the three directions are the x, y and z-axes.

To define any vector A in the Cartesian system we need the size of the components along the three axes (Ax, Ay and Az) and

three unit vectors that are parallel to the three axes (i, j and k parallel to x, y and z respectively).

  The vector A is given by

  The magnitude of A is given by

 

Non-Cartesian Systems

Although the Cartesian system is the most common one and the easiest to visualise and use, There are

two other system that are useful when considering problems with cylindrical or spherical symmetry.

In the cylindrical system the three components of the vector are defined as lying along the radial direction in the x-y plane (r), the angle between the projection onto the x-y plane and the x-axis (f) and the vertical or z component (z).

  In the spherical system the three components are along the radial direction (r), the angle between the projection onto the x-y plane and the x-axis (f) and the angle between the vector and the z-axis (q).

 

Multiplication of VectorsThere are two types of multiplication, the dot product, resulting in a scalar and the vector product resulting in a vector.

 

The dot product of two vectors

If we have two vectors A and B then the dot product of A and B is defined as

 

  where A and B are the magnitudes of vectors A and B and q is the angle between the two vectors.

  The dot product is commutative

 

  Physically the dot product represents the projection of one of the vectors on to the other times the magnitude of the other. If the two vectors are mutually perpendicular then the dot product is zero (cos90°=0).

  In Cartesian co-ordinates, if we have two vectors A and B with components (Ax, Ay, Az) and (Bx, By, Bz) respectively then

 

 

Physical application of the dot product

We know from mechanics that if a force F moves through a distance L then the work done is equal to the component of the force along the direction of movement multiplied by L. In the diagram below the component of F along the direction of movement is Fcosq. Hence the work done is FLcosq. However if we use vectors F and L to describe the force and the movement respectively we have from the definition of the dot product . Hence when a force F is moved by a distance L the work done is simply the dot product of the two vectors. This is an application of the dot product which we will

use many times in the electromagnetism course.

 

  The cross product of two vectors

If we have two vectors A and B then the cross product of A and B is defined as

  A´B = ABsinqn

  Where n is a unit vector normal to the plane containing the two vectors A and B and whose direction is given by the right-hand rule.

  The cross product of two vectors is not commutative as sin(-q)=-sinq.

A´B = - B´A

  In Cartesian co-ordinates the cross product of the vectors A and B with components (Ax, Ay, Az) and (Bx, By, Bz) is given by

  A´B = (AyBz-AzBy)i+(AzBx-AxBz)j+(AxBy-AyBx)k

 

  The cross product of a vector with itself is zero as the angle between the two vectors is 0 and sin(0)=0.

  Physical application of the cross product

A force F acts at a distance r from a point of rotation. The torque (T)about this point is the distance from where F acts to the point of

rotation (r) multiplied by the normal component of F. T=rFsinq, where q is the angle between F and the line drawn through the point of rotation. However if we define torque in terms of a vector whose magnitude gives the size of the torque and whose direction points along the axis of rotation then

T= r´F where r is the vector from the point of rotation to the point where F acts. The direction of T gives the sense of rotation from the right-hand screw rule.

 

Calculus of scalars and vectorsMuch of physics is concerned with how one quantity varies when one or more other quantities change. As in other areas of physics, in electromagnetism we will be concerned with the spatial variation (or derivative) of both scalar and vector quantities.

  The mathematics can be summarised by the use of a differential operator called 'del' (symbol Ñ) which itself has directional properties.

  In Cartesian co-ordinates Ñ is given by

  There are three physically meaningful ways in which Ñ can be applied to scalars and vectors: it can be applied to a scalar to give a vector (gradient), it can form the dot product with a vector to give a scalar (divergence) and it can form the cross product with a vector to give another vector (curl).

 

Gradient Ñ

The gradient of a scalar function f is written Ñf or grad f and is given in the Cartesian system by

 

  The resultant quantity is a vector.

  e.g. if f(x,y,z)=2x2+y3+z2xy then Ñf=i(4x+z2y)+j(3y2+z2x)+k2xyz

  Physical significance of the gradient. At any point the gradient of a function points in the direction corresponding to that for which the function varies most rapidly. The magnitude of the gradient vector gives the size of this maximum variation.

  Example If f(x,y) gives the height (or alternatively the z co-ordinate) of a surface as a function of the x and y co-ordinates then at any point Ñf will point in the direction of maximum slope of the surface.

A small ball placed on the surface will tend to roll along the direction opposite to Ñf. The gravitational force acting on the ball is -mgÑf where m is its mass and g is the acceleration due to gravity. The negative sign arises because the force acts in the opposite direction to Ñf. Alternatively if we define U(x,y) as the gravitational potential energy of the ball (U(x,y)=mg f(x,y)) then the gravitational force = -ÑU. This is a general result: Force = -gradient(potential energy). The potential energy may be gravitational, electrical etc.

 

Divergence Ñ×

The divergence of a vector A is written as Ñ×A or div A and is given by

 

  the resultant quantity is a scalar

  e.g. if A=3x2yzi+x2z2j+z2k then Ñ×A=6xyz+2z

  Physical significance. When the divergence of a vector is positive at a given point then there is a source of the vector field at that point. A negative divergence implies a sink for the vector field. We can hence think of the divergence of a vector as telling us how much of the vector field starts (or terminates) at a given point.

In (a) the vector has a constant magnitude so its divergence is zero. In (b) the x-component increases along the x-direction. This vector hence has a non-zero, positive divergence.

 

Curl Ñ´

The curl of a vector A is written as Ñ´A or curl A and is given by

 

=

 

the resultant is a vector

 

eg A=yzi-2x2yzj+3x2y2zk Ñ´A=(6x2yz--2x2z)i+(y-6xy2z)j+(-4xyz-z)k

 

Physical significance A non-zero curl implies that the corresponding vector field has a sort of rotational property. One way to look for a curl is to imagine that the vector field corresponds to the flow of water. If we place a small paddle wheel in the field then the presence of a non-zero curl suggests that the wheel will rotate.

 

In the above examples for (a) although the field increases along the direction in which it points it produces no rotation of the wheel. However in (b) the field points along x but increases along the y-axis and hence produces a rotation of the wheel. Hence the curl is related to how the field changes as we move across the field. This can also be seen because the expression for curl contains terms ¶Ax/¶y etc.

To some extent curl and div are complementary. The latter requires that the field increases when moving along the field direction, the former that the field increases when moving across the field direction.

 

Relationships

From the definitions of grad, div and curl the following relationships can be established

  Ñ´(Ñf)=0 the curl of a gradient is equal to zero

  Ñ×(Ñ´A)=0 the divergence of a curl is equal to zero

  Ñ×(Ñf)=Ñ2f=

this is the divergence of a gradient

 

 

 

Non-cartesian co-ordinatesAll of the previous examples are for cartesian co-ordinates. For other systems related, but different, expressions exist for grad, div and curl

  e.g. in cylindrical co-ordinates the gradient is given by

 

  For this course you do not need to remember the expressions for non-cartesian systems but you need to know how to apply them where necessary.

 

IntegrationThere are two main types of integration for vectors, line and surface

  Line integrals

The line integral of a vector A between the points a and b is given by

as we move along a path between the points a and b, at each step we take the component of A which lies along the direction we are moving (given by the vector dl) and multiply it by the distance we move through. The line integral is the sum of all these individual values as we move from a to b.

In general the path taken between the points a and b must be specified. However for a certain class of vectors the result of the integral is independent of the path taken. Such vectors are said to be conservative.

If the line integral is performed around a closed path (initial and final points are the same) then a circle is placed on the integral symbol

If the vector A represents a force then the line integral of A between two points gives the work done in moving between these two points.

 

Surface integrals

The surface integral of the vector A over the surface s is defined as

the surface is split into an infinite number of infinitesimally small sections. For each section the product of the area of the section (dS) and the component of A normal to the surface is formed. The integral is the sum of all these products. If the surface is a closed one (no edges) then a circle is placed on the integral sign

 

Relationships between integrals

 

Divergence theorem

This states

 

  in words 'The surface integral of any vector over a closed surface S is equal to the divergence of that vector integrated over the volume t enclosed by S.'

 

Stokes' theorem

This states

 

  in words 'The line integral of any vector around a closed path is equal to the surface integral of the curl of that vector integrated over a surface S which is bounded by the path of the line integral.'

Return to main page