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Electromagnetic, Electrostatic and Magnetostatic Fields
Electromagnetic fields are characterized by coupled, dynamic (time-varying) electric and magnetic fields and are governed by the complete setof Maxwell’s equations (four coupled equations). According to Maxwell’sequations, a time-varying electric field cannot exist without the asimultaneous magnetic field, and vice versa.
Under static conditions, the time-derivatives in Maxwell’s equationsgo to zero, and the set of four coupled equations reduce to two uncoupledpairs of equations. One pair of equations governs electrostatic fields whilethe other set governs magnetostatic fields. This decoupling of Maxwell’sequations illustrates that static electric fields can exist in the absence ofmagnetic fields and vice versa. Stationary charges produce electrostaticfields while magnetostatic fields are produced by steady (DC) currents orpermanent magnets.
Maxwell’s Equations(electromagnetic fields)
Maxwell’s Equations Maxwell’s Equations (electrostatic fields) (magnetostatic fields)
Electrostatic Fields
Electrostatic fields are static (time-invariant) electric fields producedby static (stationary) charges. The mathematical definition of theelectrostatic field is derived from Coulomb’s law which defines the vectorforce between two point charges.
Coulomb’s Law
1 2Given point charges [q , q (units=C)]
1 2in air located by vectors R and R ,respectively, the vector force acting on
2 1 12charge q due to q [F (units=N)] is definedby Coulomb’s law as
o1 2where is a unit vector pointing from q to q and å is the free-space
opermittivity [å = 8.854×10 F/m]. The permittivity of air is!12
approximately equal to that of free space (vacuum). Note that, accordingto Coulomb’s law, the force between the point charges is directlyproportional to the product of the charges and inversely proportional to thesquare of the separation distance between the charges. The unit vector
1 2pointing from q to q can be written as
Inserting this equation for the unit vector into Coulomb’s law gives analternative form of Coulomb’s law:
The first form of Coulomb’s law allows one to easily identify both themagnitude and direction of the vector force, while the second form does notrequire an explicit unit vector determination.
Note that the unit vector direction is defined according to whichcharge is exerting the force and which charge is experiencing the force. This convention assures that the resulting vector force always points in theappropriate direction (opposite charges attract, like charges repel).
The point charge is a mathematical approximation to a very smallvolume charge. The definition of a point charge assumes a finite chargelocated at a point (zero volume). The point charge model is applicable to small charged particles (like electrons) or when two charged bodies areseparated by such a large distance that these bodies appear as point chargesto each other.
Given multiple point charges in a region, the principle ofsuperposition is applied to determine the overall vector force on a particularcharge. The total vector force acting on the charge equals the vector sumof the individual forces.
Force Due to Multiple Point Charges (Superposition)
1Given a point charge q in the vicinity of a set of N point charges (q ,
N2q ,..., q ), the total vector force on q is the vector sum of the individualforces due to the N point charges.
N1, 2, ... , F / total vector force on q due to q q q
Electric Field
According to Coulomb’s law, the vector force between two pointcharges is directly proportional to the product of the two charges. Alternatively, we may view each point charge as producing a force fieldaround it (electric field) which acts on any charge in its vicinity. Since apoint charge will repel a charge of like sign, but attract a charge of unlikesign, we must adopt a convention as to the sign on the electric field force. We will adopt the convention that the direction of the vector electric fieldis the direction of the force on positive charge. Using a positive testcharge to measure the electric field, the electric field is defined as thevector force per unit charge experienced by the test charge.
q - point charge producing the electric field
t q - positive test charge used tomeasure the electric field of q
RN- locates the source point PN(location of source charge q)
R - locates the field point P
t(location of test charge q )
tFrom Coulomb’s law, the force on the test charge q due to the charge q is
The vector electric field produced by q at the field point P (designated asE) is found by dividing the vector force on the test charge F by the test
tcharge q .
Note that the electric field produced by q is independent of the magnitude
tof the test charge q . The electric field units [Newtons per Coulomb (N/C)]are normally expressed as Volts per meter (V/m) according to the followingequivalent relationship:
For the special case of a point charge at the origin (RN = 0), theelectric field reduces to the following spherical coordinate expression:
Note that the electric field points radially outward given a positive pointcharge at the origin and radially inward given a negative point charge at theorigin. In either case, the electric field of the a point charge at the origin isspherically symmetric and the magnitude of the electric field varies as R .!2
The electric field due to multiple point charges can be determined
tusing the principle of superposition. The vector force on a test charge q at
N N1 2 1 2R due to a system of point charges (q , q ,..., q ) at (R N, R N,..., R N) is, bysuperposition,
oOn the spherical surface S of radius R , we have
Note the outward pointing normal requirement in Gauss’s law is a directresult of our electric field (flux) convention.
By using an outward pointing normal, we obtain the correct sign on theenclosed charge.
Gauss’s law can also be used to determine the electric fields producedby simple charge distributions that exhibit special symmetry. Examples ofsuch charge distributions include uniformly charged spherical surfaces andvolumes.
R! +By symmetry, on S (and S ), D is uniform and has only an -component.
or
+Gauss’s law can be applied on S to determine the electric field outside the
ocharged sphere [E(R>R )].
or
Even integrand, Symmetric limits
(Potential in the x-y plane due to a uniform line charge of length 2a centered at the origin)
Electric Field as the Gradient of the Potential
The potential difference between two points in an electric field can bewritten as the line integral of the electric field such that
From the equation above, theincremental change in potential alongthe integral path is
where è is the angle between thedirection of the integral path and theelectric field. The derivative of thepotential with respect to position alongthe path may be written as
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