Lecture 32

Image Theory

32.1 Image Theory

Image theory can be used to derived closed form solution to boundary value problems when

the geometry is simple and has a lot of symmetry. The closed form solutions in turn offer

physical insight into the problems. This theory or method is also discussed in many textbooks

[1, 48, 59, 71, 155, 162, 163].

32.1.1 A Note on Electrostatic Shielding

For electrostatic problems, a conductive medium suffices to produce surface charges that

shield out the electric field from the conductive medium. If the electric field is not zero,

then since J = σE, the electric current inside the conductor will keep flowing until inside the

conductive medium E = 0, and no electric current can flow in the conductor. In other words,

when the field reaches the quiescent state, the charges redistribute themselves so as to shield

out the electric field, and that the total internal electric field, E = 0. And from Faraday’s

law that tangential E field is continuous, then ˆn × E = 0 on the conductor surface since

ˆn × E = 0 inside the conductor. Figure 32.1 shows the static electric field, in the quiescent

state, between two conductors (even though they are not PEC), and the electric field has to

be normal to the conductor surfaces.

32.1.2 Relaxation Time

The time it takes for the charges to move around until they reach their quiescent distribution

is called the relaxation time. It is very much similar to the RC time constant of an RC circuit

consisting of a resistor in series with a capacitor. It can be proven that this relaxation time

is related to ε/σ, but the proof is beyond the scope of this course. Note that when σ → ∞,

the relaxation time is zero. In other words, in a perfect conductor or a superconductor, the

charges can reorient themselves instantaneously if the external field is time-varying.

Electrostatic shielding or low-frequency shielding is important at low frequencies. The

Faraday cage is an important application of such a shielding.

319

320 Electromagnetic Field Theory

Figure 32.1: The objects can just be conductors, and in the quiescent state (static state), the

tangential electric field will be zero on their surfaces.

However, if the conductor charges are induced by an external electric field that is time

varying, then the charges have to constantly redistribute/re-orient themselves to try to shield

out the incident time-varying electric field. Currents have to constantly flow around the

conductor. Then the electric field cannot be zero inside the conductors as shown in Figure

32.2. In other words, a finite conductor cannot shield out completely a time-varying electric

field.

Image Theory 321

Figure 32.2: If the source that induces the charges on the conductor is time varying, the

current in the conductor is always nonzero so that the charges can move around to respond

to the external time-varying charges.

For a perfect electric conductor (PEC), E = 0 inside with the following argument: Because

J = σE where σ → ∞, let us assume an infinitesimally time-varying electric field in the PEC

to begin with. It will yield an infinite electric current, and hence an infinite time-varying

magnetic field. A infinite time-varying magnetic field in turn yields an infinite electric field

that will drive an electric current, and these fields and current will be infinitely large. This is

an unstable sequence of events if it is true. Hence, the only possibility is for the time-varying

electromagnetic fields to be zero inside a PEC.

Thus, for the PEC, the charges can re-orient themselves instantaneously on surface when

the inducing electric fields from outside are time varying. In other words, the relaxation time

ε/σ is zero. As a consequence, the time-varying electric field E is always zero inside PEC,

and hence ˆn × E = 0 on the surface of the PEC.

322 Electromagnetic Field Theory

32.1.3 Electric Charges and Electric Dipoles

Image theory for a flat conductor surface or a half-space is quite easy to derive. To see that,

we can start with electro-static theory of putting a positive charge above a flat plane. As

mentioned before, for electrostatics, the plane or half-space does not have to be a perfect

conductor, but only a conductor (or a metal). The tangential static electric field on the

surface of the conductor has to be zero.

The tangential static electric field can be canceled by putting an image charge of opposite

sign at the mirror location of the original charge. This is shown in Figure 32.3. Now we can

mentally add the total field due to these two charges. When the total static electric field due

to the original charge and image charge is sketched, it will look like that in Figure 32.4. It

is seen that the static electric field satisfies the boundary condition that ˆn × E = 0 at the

conductor interface due to symmetry.

Figure 32.3: The use of image theory to solve the BVP of a point charge on top of a conductor.

The boundary condition is that ˆn × E = 0 on the conductor surface.

Image Theory 323

Figure 32.4: The total electric of the original problem and the equivalent problem when we

add the total electric field due to the original charge and the image charge.

An electric dipole is made from a positive charge placed in close proximity to a negative

charge. Using that an electric charge reflects to an electric charge of opposite polarity above

a conductor, one can easily see that a static horizontal electric dipole reflects to a static

horizontal electric dipole of opposite polarity. By the same token, a static vertical electric

dipole reflects to static vertical electric dipole of the same polarity as shown in Figure 32.5.

324 Electromagnetic Field Theory

Figure 32.5: On a conductor surface, a horizontal static dipole reflects to one of opposite

polarity, while a static vertical dipole reflects to one of the same polarity. If the dipoles are

time-varying, then a PEC will have a same reflection rule.

If this electric dipole is a Hertzian dipole whose field is time-varying, then one needs

a PEC half-space to shield out the electric field. Also, the image charges will follow the

original dipole charges instantaneously. Then the image theory for static electric dipoles over

a half-space still holds true if the dipoles now become Hertzian dipoles.

32.1.4 Magnetic Charges and Magnetic Dipoles

A static magnetic field can penetrate a conductive medium. This is apparent from our

experience when we play with a bar magnet over a copper sheet: the magnetic field from the

magnet can still be experienced by iron filings put on the other side of the copper sheet.

However, this is not the case for a time-varying magnetic field. Inside a conductive

medium, a time-varying magnetic field will produce a time-varying electric field, which in

turn produces the conduction current via J = σE. This is termed eddy current, which by

Lenz’s law, repels the magnetic field from the conductive medium.1

Now, consider a static magnetic field penetrating into a perfect electric conductor, an

minute amount of time variation will produce an electric field, which in turn produces an

infinitely large eddy current. So the stable state for a static magnetic field inside a PEC is

for it to be expelled from the perfect electric conductor. This in fact is what we observe when

a magnetic field is brought near a superconductor. Therefore, for the static magnetic field,

where B = 0 inside the PEC, then ˆn · B = 0 on the PEC surface.

Now, assuming a magnetic monopole exists, it will reflect to itself on a PEC surface so

that ˆn · B = 0 as shown in Figure 32.6. Therefore, a magnetic charge reflects to a charge of

1The repulsive force occurs by virtue of energy conservation. Since “work done” is needed to set the eddy

current in motion, or to impart kinetic energy to the electrons forming the eddy current, a repulsive force is

felt in Lenz’s law so that work is done in pushing the magnetic field into the conductive medium.

Image Theory 325

similar polarity on the PEC surface.

Figure 32.6: On a PEC surface, ˆn · B = 0. Hence, a magnetic monopole on top of a PEC

surface will have magnetic field distributed as shown.

By extrapolating this to magnetic dipoles, they will reflect themselves to the magnetic

dipoles as shown in Figure 32.7. A horizontal magnetic dipole reflects to a horizontal magnetic

dipole of the same polarity, and a vertical magnetic dipole reflects to a vertical magnetic dipole

of opposite polarity. Hence, a dipolar bar magnet can be levitated by a superconductor when

this magnet is placed closed to it. This is also known as the Meissner effect [164], which is

shown in Figure 32.8.

A time-varying magnetic dipole can be made from a electric current loop. Over a PEC, a

time-varying magnetic dipole will reflect the same way as a static magnetic dipole as shown

in Figure 32.7.

326 Electromagnetic Field Theory

Figure 32.7: Using the rule of how magnetic monopole reflects itself on a PEC surface, the

reflection rules for magnetic dipoles can be ascertained.

Figure 32.8: On a PEC (superconducting) surface, a vertical magnetic dipole reflects to one

of opposite polarity. Hence, the dipoles repel each other displaying the Meissner effect. The

magnet, because of the repulsive force from its image, levitates above the superconductor

(courtesy of Wikipedia [165]).

Image Theory 327

32.1.5 Perfect Magnetic Conductor (PMC) Surfaces

Magnetic conductor does not come naturally in this world since there are no free-moving

magnetic charges around. Magnetic monopoles are yet to be discovered. On a PMC surface,

by duality, ˆn × H = 0. At low frequency, it can be mimicked by a high μ material. One can

see that for magnetostatics, at the interface of a high μ material and air, the magnetic flux is

approximately normal to the surface, resembling a PMC surface. High μ materials are hard

to find at higher frequencies. Since ˆn × H = 0 on such a surface, no electric current can flow

on such a surface. Hence, a PMC can be mimicked by a surface where no surface electric

current can flow. This has been achieved in microwave engineering with a mushroom surface

as shown in Figure 32.9 [166]. The mushroom structure consisting of a wire and an end-cap,

can be thought of as forming an LC tank circuit. Close to the resonance frequency of this

tank circuit, the surface of mushroom structures essentially becomes open circuits resembling

a PMC. Therefore, there is no surface electric current on this surface, and the tangential

magnetic field is small, the hallmark of a good magnetic conductor.

Figure 32.9: A mushroom structure operates like an LC tank circuit. At the right frequency,

the surface resembles an open-circuit surface where no current can flow. Hence, tangential

magnetic field is zero resembling perfect magnetic conductor (courtesy of Sievenpiper [166]).

Mathematically, a surface that is dual to the PEC surface is the perfect magnetic conductor

328 Electromagnetic Field Theory

(PMC) surface. The magnetic dipole is also dual to the electric dipole. Thus, over a PMC

surface, these electric and magnetic dipoles will reflect differently as shown in Figure 32.10.

One can go through Gedanken experiments and verify that the reflection rules are as shown

in the figure.

Figure 32.10: Reflection rules for electric and magnetic dipoles over a PMC surface.

Figure 32.11: Image theory for multiple images [29].

32.1.6 Multiple Images

For the geometry shown in Figure 32.11, one can start with electrostatic theory, and convince

oneself that ˆn × E = 0 on the metal surface with the placement of charges as shown. For

conducting media, they charges will relax to the quiescent distribution after the relaxation

time. For PEC surfaces, one can extend these cases to time-varying dipoles because the

Image Theory 329

charges in the PEC medium can re-orient instantaneously (i.e. with zero relaxation time) to

shield out or expel the E and H fields. Again, one can repeat the above exercise for magnetic

charges, magnetic dipoles, and PMC surfaces.

Figure 32.12: Image theory for a point charge near a cylinder or a sphere can be found in

closed form [29].

32.1.7 Some Special Cases

One curious case is for a static charge placed near a conductive sphere (or cylinder) as shown in

Figure 32.12.2 A charge of +Q reflects to a charge of −QI inside the sphere. For electrostatics,

the sphere (or cylinder) need only be a conductor. However, this cannot be generalized to

electrodynamics or a time-varying problem, because of the retardation effect: A time-varying

dipole or charge will be felt at different points asymmetrically on the surface of the sphere

from the original and image charges. Exact cancelation of the tangential electric field cannot

occur for time-varying field.

Figure 32.13: A static charge over a dielectric interface can be found in closed form.

When a static charge is placed over a dielectric interface, image theory can be used to

find the closed form solution. This solution can be derived using Fourier transform technique

2This is worked out in p. 48 and p. 49, Ramo et al [29].

330 Electromagnetic Field Theory

which we shall learn later [34]. It can also be extended to multiple interfaces. But image

theory cannot be used for the electrodynamic case due to the different speed of light in

different media, giving rise to different retardation effects.

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