ELECTROMAGNETIC FIELDS

Maxwell's Fourth Equation.

Capacitance. If the terminals of the dry cell are connected by wires to a pair of parallel plates, the plates become charged in a small fraction of second and then a potential difference of 1.5 volts appears between the two plates. This device consisting of parallel plates separated by air or some insulating material is a simple form a capacitor. Each of the two plates has an electric charge Q, but of opposite sign. This charge Q is directly proportional to the impressed potential difference V. That is,
Q = CV, (14)
where the constant of proportionality C is called the capacitance of the capacitor. The symbol for capacitance is C and solving formula (14) for the capacitance, we find that the capacitance of a capacitor is the ratio of its charge to the potential difference across its plates. That is,
C = Q/V. (15)
The unit of capacitance is the farad, abbreviated with letter f, and is equal to a coulomb per volt. The farad is so large that, for practical purposes, capacitance is usually measured in microfarads (μf) or micromicrofarads (μμf) that is also called a picofarads (pf), whose values are
1 μf = 10^-6, and 1 μμf = 1 pf = 10^-12 f.
In general a capacitor is a device that stores electrical potential energy in the form of an electric field. We can calculate the electric potential energy U of a charged capacitor by computing the amount of work W that must be performed to charge the capacitor to an electric potential V. Using the definition of electric potential as the instantaneous rate that work is done per unit of charge, that is,
V = dW/dq, (16)
and, solving equation (16) for dW, the work done in charging the capacitor from q = 0 to q = Q is found by integrating the differential equation
dW = Vdq = (q/C)(dq) = qdq/C.
That is,
U = W_0→Q = ∫₀^QdW = ∫₀^QVdq = ∫₀^Qqdq/C = ½Q²/C,
since at any time when the charge q is being transferred, the potential difference V between the plates is
V = q/C.
Thus the electric potential energy U of a capacitor of capacitance C whose plates have the charge Q is
U = ½Q²/C, (17)
or, using formula (14), in terms of capacitance and voltage,
U = ½[(CV)²/C] = ½CV², (18)
or in terms of charge and voltage,
U = ½[(CV)V] = ½QV, (19)
where the capacitor has electric potential energy U because work was done to charge the capacitor. This potential electric energy is stored in the electric field between the plates of the capacitor and the density of this potential electric energy can be calculated. Energy density u is defined as the energy per unit volume. For a parallel plate capacitor (with small plate separation and large plate area), since the capacitance
C = εA/s
and the potential difference across its plates V = Es, the electric potential energy is
U = ½CV² = ½(εA/s)(Es)² = ½εE²(As),
where the factor As is the volume occupied by the electric field between the plates of the capacitor. Thus the electric energy density is
u = U/As = [½εE²(As)]/(As) = ½εE². (20)
That is, the electric energy density of an electric field is directly proportionality to the square of the electric field intensity. While this relation is derived for the specific case of a parallel plate capacitor, it is a completely general result.

Parallel Capacitance. If two capacitors are connected in parallel, then their total capacitance is
C = Q/V = (Q₁ + Q₂)/V = Q₁/V + Q₂/V = C₁ + C₂. Thus,
C = C₁ + C₂. (21)
That is, the total capacitance of a parallel arrangement of capacitance elements is equal to the sum of the capacitances of each element.

Series Capacitance. If two capacitors are connected in series, then the total capacitance is
Q/C = V = V₁ + V₂ = Q₁/C₁ + Q₂/C₂.
But since Q = Q₁ = Q₂, then
1/C = 1/C₁ + 1/C₂. (22)
That is, the reciprocal of the total capacitance of a series arrangement of the capacitance elements is equal to the sum of the reciprocals of the capacitance of each element in the series.

Charging a Capacitor. Now consider a circuit consisting of a capacitor with a capacitance C in series with a resistance R connected through a switch to a battery with an EMF of V volts. Assume that the capacitor is initially uncharged (the charge on the two plates are equal) and there is no current flowing in the circuit when the switch is open. When the switch is closed at time t = 0, charges begin to flow, setting up a current in the circuit and the capacitor begins to charge. Since the plates of the capacitor are separated by an insulator of some kind so that no current flows between the plates. As the charge builds up on the plate connected to the positive terminal of the battery, the charge on the other plate of the capacitor connected to the negative terminal of the battery is repelled as the current flows toward the negative terminal of the battery. This continues until the capacitor is fully charged. The value of the maximum charge depends upon the EMF of the battery. Once the maximum charge is reached, the current in circuit is zero.

To analyze quantitatively what happens after the switch is closed, let us start with the fact that the EMF voltage of the battery is equal to the sum of potential differences across the capacitor
V_C = q/C and across the resistor
V_R = IR. That is,
V = V_R + V_C = IR + q/C. (23)
Note that the charge and current are variables whose instantaneous values change as the capacitor charges. At time t = 0, when the switch closed, the charge on the capacitor is zero, and the initial value of current I₀ is a maximum and is equal to I₀ = V/R. At this time the applied voltage is entirely across the resistor. Later, when the capacitor is fully charged to its maximum value Q, the charges cease to flow (the current is zero), and the applied voltage is entirely across the capacitor. The charge Q on the capacitor is then equal CV. To determine an analytical expression for the time dependence of the charge and current, the equation (23) must be differentiation with respect to time.
dV/dt = R(dI/dt) + (1/C)(dq/dt),
since I = dq/dt.
Since the EMF voltage is zero, then dV/dt = 0. Thus
0 = R(dI/dt) + (1/C)(dq/dt) or
R(dI/dt) = -(1/C)I,
Separating the variables and rearranging the equation, we get
dI/I = (1/RC)dt.
Since R and C are constants, this can be integrated using the initial conditions that at t = 0,
I = I₀. Integrating, we get
ln(I/I₀) = -t/RC. (24)
Solving this equation for I, we get a formula for current as function of time.
I(t) = I₀e^-t/RC = (V/R)e^-t/RC, (25)
where e is the natural logarithmic base and the I₀ = V/R is the initial current.

In order to find the charge on the capacitor as a function of time, substitute I = dq/dt into equation (25),
I(t) = dq/dt = (V/R)e^-t/RC
or, separating the variables, .
dq = (V/R)e^-t/RCdt
Integrating, using the condition q = 0 at t = 0, we get
q(t) = CV[1 - e^-t/RC] = Q[1 - e^-t/RC], (26)
since Q = CV is the maximum charge on the capacitor.

Note that the charge is zero at t = 0 and approaches exponentially the maximum value of Q at time t . Also the current has its maximum value I₀ = V/R at t = 0 and decays exponentially to zero as t → ∞. The quantity RC, which appears in the exponentials of equations (25) and (26), is called the time constant τ of the circuit. It represent the time it takes for the current to decrease to 1/e of its initial value; that is, in the time equal to the time constant τ, the current decays to
I = I₀e^-1 = 0.37I₀,
and the charge on the capacitor will increase to
q = Q[1 - e^-1] = Q[1 - 0.37] = 0.63Q.
The following dimensional analysis shows that the product RC has the unit of time
[τ] = [RC] = [(V/I) × (Q/V)] = [Q/I] = [Q/(Q/T)] = [Q].

Discharging a Capacitor. Now consider a circuit consisting of a capacitor C with an initial charge Q, a resistor R, and switch, but no source of EMF. When the switch is open, there is a potential difference of
V = Q/C across the capacitor and zero potential difference across the resistor since I = 0. When the switch is closed at t = 0, the capacitor begins to discharge through the resistor. At some time during the discharge, the current in the circuit is I and the charge on the capacitor is q. Now the voltage across the resistor, IR, must equal the potential difference across the capacitor, q/C.
V_R = V_C, or
IR = q/C.
However, the current in the circuit must equal the rate of decrease of charge on the capacitor; that is,
I = -dq/dt.
-R(dq/dt) = (q/C)
or, separating the variables and rearranging the equation, we get
ln(q/Q) = -(t/RC).
Integrating this equation using the fact that q = Q at t = 0, we get
q(t) = Qe^-t/RC. (27)
In order to get the current as a function of time, differentiate equation (27).
I = dq/dt = (Q/RC)e^-t/RC, (28)
where the initial current I₀ = Q/RC.
Both the charge on the capacitor and the current decay are exponential at a rate characterized by the time constant τ = RC.

Current Density. If the current is distributed uniformly throughout the cross section area of a wire, then the current density J is uniform and is given by the total current I divided by the cross-sectional area A of the wire. That is,
J = I/A, (29)
where in rationalized MKS system of units the current is measured in amperes, the cross-sectional area in square meters, and the current density in amperes per square meter (Am^-2).

If the current is not uniformly distributed, equation (29) gives the average current density. However, it is often of necessary to consider the current density at a point. This instantaneous current density is defined as the limit of the ratio of current I through a small cross-sectional area s to that area s as s approaches zero. Hence, the current density at a point is given by
J = lim(ΔI/Δs) = dI/ds, (30)
where the surface Δs is taken as normal to the current direction. The current density J is a vector point function having a magnitude equal to the current density at the point and its direction is the direction of the current at the point.

Now consider a block of the conducting material. Imagine a small rectangular cell of length L and cross-sectional area A enclosing a point P in the interior of the block with the cross-sectional area A perpendicular to the direction of the current density J. Applying Ohm's Law to this cell, we have V = IR, where V is the potential difference between the ends of the cell and R is the resistance of the conducting material in the cell. But V = Es = EL and by equation (29)
I = JA; so that by substitution into Ohm's Law we get
EL = JAR or
J = (L/RA)E = σE, (31)
where σ = (L/RA) is called the conductivity of the conducting material and is the reciprocal of resistivity ρ; that is, σ = 1/ρ, where ρ = RA/L.
By making the cell enclosing point P as small as we wish, this relation can be made to apply at the point P and can be written as a vector equation.
J = σE. (32)
Equation (32) is Ohm's Law at a point and relates the current density J at a point to the electric field intensity E at the point and the conductivity σ of the material. Here it is assumed that the conducting material is homogeneous (same material throughout), isotropic (resistance between opposite faces of a cube independent of the pair chosen), and linear (resistance independent of current).

Displacement Current. Consider a voltage applied to a resistor and a capacitor in parallel. The nature of the current flow through the resistor is different from that through the capacitor. A constant voltage applied across a resistor produces a continuous flow of current of a constant value. But there will be current through the capacitor only while the voltage is changing, during charging or discharging of the capacitor.

For a voltage V across a resistor of resistance R and a capacitor of capacitance C in parallel, the current through the resistor is given by I₁ = V/R and a current through the capacitor given by
I₂ = dQ/dt = C(dV/dt), (33)
where instantaneous charge dQ in the capacitor is given by dQ = CdV. The current through the resistor is called conduction current, while the current "through" the capacitor may be called a displacement current. Although the current does not flow through the capacitor, the external effect is as though it did, since as much current flows out of one plate as flows into the opposite one. Fringing of the electric field can be neglected. Now inside each element, the electric field E equals the voltage V across the element divided by its length L. That is, E = V/L. From equation (29) the current density J₁ inside the resistor equals to I₁ divided by the cross-sectional area A; it is also equal to the product of the electric field E and the conductivity σ of the medium inside the resistor element; That is,
J₁ = I₁/A = (V/R)(1/A) = (V/L)(L/RA) = Eσ. (34)
In RMKS units, the dimensional analysis of equation (34) is
ampers/meter² = (volts/meter)(meter/ohms·meter²).
The capacitance of a parallel plate capacitor is C = εA/s, where A is the area of one of the plates and s is the spacing between them. Substituting this value for C into equation (33), we get
I₂ = dQ/dt = C(dV/dt) = (εA/s)(dV/dt) = (εA/s)(sdE/dt) = εA(dE/dt), (35)
since V = Es. Dividing equation (35) by the cross-sectional area A, we get the relation between the current density J₂ inside the capacitor that is equal to the permittivity of the nonconducting medium filling the space between the plates of the capacitor multiplied times the rate of change of the electric field intensity. That is,
I₂/A = J₂ = ε(dE/dt). (36)
In RMKS units, the dimensional analysis of equation (36) is
amperes/meter² = (farad/meter)[(volt/meter)/second] =
[(coulomb/volt)/meter][(volt/meter)/second] = (coulomb/second)(1/meter²).

Since electric flux density D = εE, then equation (36) becomes
J₂ = dD/dt. (37)
In our circuit above J₁ is called the conduction current density J_cond and J₂ is called the displacement current density J_disp. Now the total current density J in this parallel circuit is equal to the sum of J_cond and J_disp. Now, since current density J, the electric flux density D, and the electric field intensity E are actually space vectors, which all have the same direction in isotropic media, then equations (34) and (37) may be expressed more generally as vector equations.
J_cond = σE, (38)
J_disp = dD/dt = εdE/dt, (39)
and
J = J_cond + J_disp. (40)
The concept of displacement current, or displacement current density, was introduced by James Clerk Maxwell to account for the production of magnetic fields in empty space. There the conduction current is zero, and magnetic fields are due entirely to displacement currents.

Ampere's Law and Maxwell's Equation. According to Ampere's Law, the line integral of magnetic field intensity H around a closed contour or path is equal to the current I enclosed, that is,
∫_λ(H · dλ = I.
Where both conduction and displacement currents are present, the current I is the total current. If we let S be any surface whose periphery is the path of integration λ, then in terms of the current density J (the current per unit area), this total current through S is
I = ∫∫_SJ · dS.
Hence, we get
∫_λ(H · dλ = ∫∫_SJ · dS. (41)
But since by equation (40) J = J_cond + J_disp, then equation (41) may written
∫_λ(H · dλ = ∫∫_S(J_cond + J_disp) · dS. (42)
Using equation (39), equation (42) is usually written
∫_λ(H · dλ = ∫∫_S[J + (dD/dt)] · dS, (43)
where J without the subscript refers only to the conduction current density. This equation is the complete expression of the integral form of Maxwell's equation derived from Ampere's Law. By the application of Stokes' theorem to equation (43), we get the differential form of Maxwell's equation derived from Ampere's Law, sometimes also called Maxwell's postulate:
∇ × H = J + dD/dt. (44)
This equation is consistent with the equation of continuity.

Equation of Continuity. The equation of continuity follows from the principle of conservation of charge which states simply that electric charge cannot be created or destroyed, that is, electric charge can be produced only in pairs of equally strong positive and negative charges. The experimental proof, due to Franklin, is a simple experiment: Wrap an uncharged silk cloth around an uncharged glass rod. Rub the rod and separate the cloth from the rod; each individually exerts a strong electric forcs of attraction or repulsion on some third object, such as the charged leaf of tinfoil in an electroscope. Now if the charged cloth is wrap again around the charged rod, then electric forces exerted on some third object disappear. The conclusion from this experiment is that electric charge is not created or destroyed but that the rubbing separated equal amounts of charge but with opposite signs on the rod (positive) and the cloth (negative), and when they are brought together again, the equal but opposite charges cancel each other. No net charge (execss of + and -) is created by the friction of the rubbing. This is the principle of the conservation of electric charge and the equation of continuity follows from this principle, when we consider any region bounded by a closed surface. The electric current flowing through the closed surface is
I = ∫∫_SJ · dS
and this outward flow of positive charge must be balanced by a decrease of positive charge (or perhaps an increase of negative charge) within the closed surface. If the charge inside the closed surface is designated by Q, then the rate of decrease is -dQ/dt and the principle of the conservation of charge requires that
I = ∫∫_SJ · dS = -dQ/dt. (45)
Why the negative sign? The answer is that the presence or absence of a negative sign depends upon what current and charge is being considered. In circuit theory the current flow into one terminal of a capacitor is associated with the time rate of increase of charge on that plate. The current in the above equation (45) is an outward-flowing current and the time-rate of change of charge through the surface is negative, a decrease of charge within the enclosing surface.

Equation (45) is the integral form of the equation of continuity, and the differential, or point, form may be obtained by changing the surface integral into a volume integral by the divergence theorm,
∫∫_SJ · dS = ∫∫∫(∇ · J)dV,
and by representing the enclosed charge Q by the volume integral of the charge density ρ, that is,
Q = ∫∫∫ρdV.
Hence, equation (45) becomes
∫∫∫(∇ · J)dV = -d/dt∫∫∫ρdV.
If the surface is kept constant, the derivative becomes a partial derivative within the integral, that is,
∫∫∫(∇ · J)dV = ∫∫∫(-dρ/dt)dV.
Since both volume integrals are taken throughout the same arbitrary volume, then the integrands may be equated, and we get the differential or point form of the equation of continuity,
∇ · J = -(∂ρ/∂t). (46)

Maxwell's Postulate. Maxwell used Faraday's experimental law to obtain one of his four equations of electromagnetic fields in differential from:
∇ × E = -(∂B/∂t). (47)
This equation states that a time-changing magnetic field produces an electric field. According to the definition of curl, the electric field has the special property of circulation; its line integral about a closed path is not zero. To Maxwell this equation suggested that a time-changing electric field will produce a magnetic field. Taking his start from the point form of Ampere's Circuital Law,
∇ × H = J, (48)
which applies to steady magnetic field, Maxwell showed its inadequacy for time-varying electric fields by taking the divergence of each side of the equation:
∇ · ∇ × H = 0 = ∇ · J.
Since the divergence of the curl is identically zero, ∇ · J is also zero. However, according to the equation of continuity, equation (46), ∇ · J is not zero and equation (48), Ampere's Circuital Law, cannot be true for time-varying electric fields. In order to remove this unrealistic limitation, equation (48) must be amended in order for it to apply to time-varying electric fields. This can be done in the following way; let us add to the right side of equation (48) an unknown term G:
∇ × H = J + G.
Again, taking the divergence of both sides of this equation, we get
∇ · ∇ × H = 0 = ∇ · J + ∇ · G,
or,
∇ · J = -∇ · G.
But since by the equation of continuity, equation (46),
∇ · J = -(∂ρ/∂t), (46)
then
∇ · G = ∂ρ/∂t.
Using Maxwell's first equation derived from Coulomb's Law,
∇ · D = ρ and substituting
∇ · D for ρ, we get
∇ · G = ∂(∇ · D)/∂t = ∇ · ∂D/∂t.
Hence, we get the simplest solution for G:
G = ∂D/∂t,
and Ampere's circuital law in point form becomes
∇ × H = J + ∂D/∂t, (44)
which is called Maxwell's Postulate and is derived from Ampere's circuital law. This equation is consistent with the equation of continuity, Coulomb's law, and Ampere's law. [Equation (44) reduces to equation (48) for a steady state]. In general, then, for a nonsteady state, equation (44) takes the place of equation (48), and the equation ∇ × E = 0 takes the place of equation (47).

Maxwell's Postulate is not the only possible way around the inconsistency of equation (48) with the equation of continuity, equation (46). For example, let
J + ∂D/∂t = ∇ × (B + ∂G/∂t)/μ₀
with G as an arbitrary function, would work mathematically. But, Maxwell would not have been able to derive his theory of electromagnetic waves. Only with his postulate was Maxwell able to derive his theory. The exeperimental observation of such waves, with the properties predicted, may be consider the experimental basis for Maxwell's postulate. And also it can be shown that this postulate has a reasonable physical interpretation.

The Physical Interpretation of Maxwell's Postulate. Maxwell originally tried to explain his postulate on mechanical basis, but it was puzzling. An explanation can be given from another point of view.
Let us start with equation (44) and form the surface integral of ∇ × H over the surface S whose periphery is λ. We get then
∫∫_S(∇ × H) · dS = ∫∫_S(J + ∂D/∂t) · dS.
Applying Stokes' Theorem to the left side of this equation, we get
∫_λH · dλ = ∫∫_S(J + ∂D/∂t) · dS. (49)
But just as ∫_λE · dλ was called the electromotive force (emf) around the path λ, the left side of equation (49) may be defined as the magnetimotive force (mmf) around a path λ by the following formula:
mmf = ∫_λH · dλ. (50)
Substituting this into equation (49), we get
mmf = ∫∫_S(J + ∂D/∂t) · dS. (51)
Equations (49) and (51) show us that there are two ways of producing a magnetic field intensity:
(a) by means of the ordinary current density J,
as was observed by Oersted and postulated by Ampere,
(b) by means of a changing electric displacement, as postulated by Maxwell.
Since D ∝ E for air or a vacuum, then a changing electric field produces a magnetic field. This is the converse of Faraday's discovery that a changing magnetic field produces an electric field.

Consider the case where J = 0 everywhere, equation (49) becomes
∫_λH · dλ = ∫∫_S(∂D/∂t) · dS, (J = 0) (52)
which is an equivalent statement of Maxwell's postulate expressed soley in terms Maxwell's new contribution, the ∂D/∂t term.

Direct observation of a magnetic field produced by a changing electric field is difficult. We cannot look for an induced "magnetic current" because there are neither free magnetic poles nor conductors for magnetic currents. Also, a constant value of ∂D/∂t cannot be maintained long enough to measure the resulting magnetic field, as can be done for a constant electric current producing a magnetic field. Maxwell's Postulate is not as suseptible to direct experimental verification as is Faraday's Law. This is possibly the reason that it was the last fundamental law of classical electromagnetism to be discovered.

If Maxwell's postulate is the converse of Faraday's Law, then why is there not a converse to Ampere's Law? Or, putting the question differently, why does the equation (51) for mmf have two terms, while the corresponding equation for emf,
emf = -∫∫_S(∂B/∂t) · dS, (53)
has only one term? The answer is that the term missing in equation (53) would involve a current density of "magnetic current" or a flow of magnetic poles of one sign. And as has been pointed out previously, isolated or monopoles of one sign and that "magnetic currents" due to them are not found in our physical world. Thus there is no term analogous to J in equation (51) and the converse of Ampere's law does not exist. The fundamental role of electric charges leads to a certain lack of symmetry in our equations of electromagnetism.