Displacement Current

Michael Faraday revolutionized physics in 1830 by showing that electricity and magnetism were interrelated phenomena. (See Section 2.3.1.) He achieved this breakthrough by careful experimentation. Between 1864 and 1873, James Clerk Maxwell achieved a similar breakthrough by pure thought. Of course, this was only possible because he was able to take the previous experimental results of Coulomb, Ampère, Faraday, et cetera, as his starting point.

Prior to 1864, the laws of electromagnetism were written in integral form. Thus, Gauss's law (in SI units) was expressed as follows; the flux of the electric field, ${\bf E}({\bf r},t)$, through a closed surface, $S$, enclosing a volume, $V$, is equal to the net enclosed electric charge, divided by $\epsilon_0$; or

$$\oint_S {\bf E}\cdot d{\bf S} = \frac{1}{\epsilon}_0\int_V \rho({\bf r},t)\,dV,$$ (2.462)

where $\rho({\bf r},t)$ is the electric charge density. (See Section 2.1.6.) The no magnetic monopole law was expressed as follows; the flux of the magnetic field, ${\bf B}({\bf r},t)$, through any closed surface, $S$ is zero; or

$$\oint_S {\bf B}\cdot d{\bf S} = 0.$$ (2.463)

(See Section 2.2.9.) Faraday's law of electromagnetic induction was expressed as follows; the line integral of the electric field around a closed loop, $C$, is equal to minus the rate of change of the magnetic flux passing through any surface, $S$, attached to the loop; or

$$\oint_C {\bf E}\cdot d{\bf r} = -\frac{\partial}{\partial t}\int_S {\bf B}\cdot d{\bf S}.$$ (2.464)

(See Section 2.3.1.) Finally, Ampère's circuital law was expressed as follows; the line integral of the magnetic field around a closed loop $C$ is equal the net current passing through any surface, $S$, attached to the loop, multiplied by $\mu_0$; or

$$\oint_C {\bf B}\cdot d{\bf r} = \mu_0\int_S {\bf j}({\bf r},t)\cdot d{\bf S},$$ (2.465)

where ${\bf j}({\bf r},t)$ is the electric current density. (See Section 2.2.10.)

Maxwell's first great achievement was to realize that, with the aid of the divergence theorem and the curl theorem (see Sections A.20 and A.22), these laws could be re-expressed as a set of first-order partial differential equations. Of course, he wrote his equations out in component form, because modern vector notation did not come into vogue until about the time of the First World War. In modern notation, Maxwell first wrote:

$$\nabla\cdot{\bf E} = \frac{\rho}{\epsilon_0},$$ (2.466)

$$\nabla\cdot{\bf B} =0,$$ (2.467)

$$\nabla\times{\bf E} = - \frac{\partial {\bf B}}{\partial t} ,$$ (2.468)

$$\nabla\times{\bf B} = \mu_0 \,{\bf j}.$$ (2.469)

[See Equations (2.54), (2.263), (2.286), and (2.271).] Maxwell's second great achievement was to realize that these equations are not mathematically self-consistent.

Consider the integral form of Equation (2.469):

$$\oint_C {\bf B}\cdot d{\bf r} = \mu_0\int_S{\bf j}\cdot d{\bf S}.$$ (2.470)

This equation states that the line integral of the magnetic field around a closed loop $C$ is equal to the flux of the current density through the loop, multiplied by $\mu_0$. The problem is that the flux of the current density through a loop is not, in general, a well-defined quantity. In order for the flux to be well defined, the integral of ${\bf j}\cdot d{\bf S}$ over some surface $S$ attached to a loop $C$ must depend on $C$, but not on the details of $S$. This is only the case if

$$\nabla \cdot {\bf j} = 0.$$ (2.471)

(See Section A.20.) Unfortunately, the previous condition is only satisfied for non-time-varying fields.

Why do we say that, in general, $\nabla\cdot{\bf j} \neq 0$? Consider the flux of ${\bf j}$ out of some closed surface, $S$, enclosing a volume, $V$. This is clearly equivalent to the instantaneous rate at which electric charge flows out of $S$. However, because electric charge is a conserved quantity (see Section 2.1.2), the rate at which charge flows out of $S$ must equal the rate of decrease of the charge contained in volume $V$. Thus,

$$\oint_S {\bf j} \cdot d{\bf S} = - \frac{\partial}{\partial t}\! \int_V \rho\,dV.$$ (2.472)

Making use of the divergence theorem (see Section A.20), the previous equation yields

$$\nabla\cdot{\bf j} = -\frac{\partial \rho}{\partial t}.$$ (2.473)

Thus, it is only the case that $\nabla\cdot{\bf j} =0$ in a steady state situation; that is, when $\partial/\partial t \equiv 0$.

The problem with Ampère's circuital law is well illustrated by the following very famous example. Consider a long straight wire interrupted by a parallel plate capacitor. Suppose that $C$ is some loop that circles the wire. In the time-independent case, the capacitor acts like a break in the wire, so no current flows, and no magnetic field is generated. There is clearly no problem with Ampère's circuital law in this case. However, in the time-dependent case, a transient current flows in the wire as the capacitor charges up, or charges down, and so a transient magnetic field is generated. Thus, the line integral of the magnetic field around $C$ is (transiently) non-zero. According to Ampère's circuital law, the flux of the current density through any surface attached to $C$ should also be (transiently) non-zero. Let us consider two such surfaces. The first surface, $S_1$, intersects the wire. See Figure 2.39. This surface causes us no problem, because the flux of ${\bf j}$ though the surface is clearly non-zero (because the surface intersects a current-carrying wire). The second surface, $S_2$, passes between the plates of the capacitor, and, therefore, does not intersect the wire at all. Clearly, the flux of the current density through this surface is zero. The current density fluxes through surfaces $S_1$ and $S_2$ are obviously different. However, both surfaces are attached to the same loop $C$, so the fluxes should be the same, according to Ampère's circuital law, (2.470). Note, however, that although the surface $S_2$ does not intersect any electric current, it does pass through a region containing a strong, time-varying electric field, as it threads between the plates of the charging (or discharging) capacitor. Perhaps, if we add a term involving $\partial {\bf E}/{\partial t}$ to the right-hand side of Equation (2.469) then we can somehow fix up Ampère's circuital law? This is, essentially, how Maxwell reasoned one hundred and fifty years ago.

Figure 2.39: Application of Ampère's circuital law to a charging, or discharging, capacitor.

Let us try out this scheme. Suppose that we write

$$\nabla\times{\bf B} = \mu_0\, {\bf j} +\lambda\,\frac{\partial {\bf E}}{\partial t},$$ (2.474)

instead of Equation (2.469). Here, $\lambda$ is some constant. Does this resolve our problem? We require the flux of the right-hand side of the previous equation through some loop $C$ to be well defined; that is, the flux should only depend on $C$, and not the particular surface $S$ (which spans $C$) upon which it is evaluated. This is another way of saying that we require the divergence of the right-hand side of the previous equation to be zero. (See Section A.20.) In fact, we can see that this is necessary for mathematical self-consistency, because the divergence of the left-hand side is identically zero. (See Section A.22.) So, taking the divergence of Equation (2.474), we obtain

$$0= \mu_0 \,\nabla\cdot{\bf j} +\lambda \,\frac{\partial\, \nabla\cdot{\bf E}}{\partial t}.$$ (2.475)

But, we know that

$$\nabla\cdot {\bf E} = \frac{\rho}{\epsilon_0}$$ (2.476)

[see Equation (2.466)], so combining the previous two equations we arrive at

$$\mu_0\, \nabla\cdot{\bf j} + \frac{\lambda}{\epsilon_0}\, \frac{\partial\rho}{\partial t} =0.$$ (2.477)

Now, our charge conservation law, (2.473), can be written

$$\nabla\cdot{\bf j} +\frac{\partial\rho}{\partial t} = 0.$$ (2.478)

The previous two equations are in agreement provided $\lambda = \epsilon_0\,\mu_0$. So, if we modify Equation (2.469) such that it reads

$$\nabla\times{\bf B} = \mu_0 \,({\bf j} +{\bf j}_d),$$ (2.479)

where

$${\bf j}_d = \epsilon_0\,\frac{\partial{\bf E}}{\partial t},$$ (2.480)

then we find that the divergence of the right-hand side is zero, as a consequence of charge conservation. The additional term, ${\bf j}_d$, is known as the displacement current density (this name was invented by Maxwell). In summary, we have shown that, although the flux of the real current density through a loop is not well defined, if we form the sum of the real current density and the displacement current density then the flux of this new quantity through a loop is well defined.

Of course, the displacement current is not a current at all. It is, in fact, associated with the induction of magnetic fields by time-varying electric fields. Maxwell came up with this rather curious name because many of his ideas regarding electric and magnetic fields were completely wrong. For instance, Maxwell believed in the aether (a tenuous invisible medium permeating all space; see Section 3.1.2), and he thought that electric and magnetic fields corresponded to stresses in this medium. He also thought that the displacement current was associated with a displacement of the aether (hence, the name). The reason that these misconceptions did not invalidate Maxwell's equations is quite simple. Maxwell based his equations on the results of experiments, and he added in his extra term so as to make these equations mathematically self-consistent. Both of these steps are valid irrespective of the existence or non-existence of the aether.

The field equations (2.466)–(2.469) are derived directly from the results of famous nineteenth century experiments. So, if a new term involving the time derivative of the electric field needs to be added to one of these equations, for the sake of mathematical consistency, why is there is no corresponding nineteenth century experimental result that demonstrates this fact? Actually, as is described in the following, the new term corresponds to an effect that is far too small to have been observed in the nineteenth century.

First, we shall show that it is comparatively easy to detect the induction of an electric field by a changing magnetic field in a desktop laboratory experiment. The Earth's magnetic field is about 1 gauss (that is, $10^{-4}$ tesla). Magnetic fields generated by electromagnets (that will fit on a laboratory desktop) are typically about one hundred times larger than this. Let us, therefore, consider a hypothetical experiment in which a 100 gauss magnetic field is switched on suddenly. Suppose that the field ramps up in one tenth of a second. What electromotive force is generated in a 10 centimeter square loop of wire located in this field? Faraday's law is written

$$V = -\frac{\partial}{\partial t} \oint {\bf B}\cdot d{\bf S} \simeq \frac{ B\,A}{t},$$ (2.481)

where $B=0.01$ tesla is the magnetic field-strength, $A=0.01$ m$^2$ the area of the loop, and $t=0.1$ seconds the ramp time. (See Section 2.3.1.) It follows that $V \simeq 1$ millivolt, which is easily detectable. In fact, most hand-held laboratory voltmeters are calibrated in millivolts. It is, thus, clear that we would have no difficulty whatsoever detecting the magnetic induction of electric fields in a nineteenth-century-style laboratory experiment.

Let us now consider the electric induction of magnetic fields. Suppose that our electric field is generated by a parallel plate capacitor of spacing one centimeter that is charged up to $100$ volts. This gives an electric field of $10^4$ volts per meter. Suppose, further, that the capacitor is discharged in one tenth of a second. The law of electric induction is obtained by integrating Equation (2.479), and neglecting the first term on the right-hand side. Thus,

$$\oint_C {\bf B} \cdot d{\bf r} = \epsilon_0\, \mu_0 \,\frac{\partial}{\partial t}\! \int_S {\bf E} \cdot d{\bf S}.$$ (2.482)

Let us consider a loop that is 10 centimeters square. What is the magnetic field generated around this loop (which we could try to measure with a Hall probe)? Very approximately, we find that

$$l \,B \simeq \epsilon_0\, \mu_0 \,\frac{ E\, l^2}{t},$$ (2.483)

where $l=0.1$ meters is the dimensions of the loop, $B$ the magnetic field-strength, $E=10^4$ volts per meter the electric field, and $t=0.1$ seconds the decay time of the field. We obtain $B\simeq 10^{-9}$ gauss. Modern technology is unable to detect such a small magnetic field, so we cannot really blame nineteenth century physicists for not discovering electric induction experimentally.

Note, however, that the displacement current is detectable in some modern experiments. Suppose that we take an FM radio signal, amplify it so that its peak voltage is one hundred volts, and then apply it to the parallel plate capacitor in the previous hypothetical experiment. What size of magnetic field would this generate? A typical FM signal oscillates at $10^9$ Hz, so $t$ in the previous example changes from $0.1$ seconds to $10^{-9}$ seconds. Thus, the induced magnetic field is about $10^{-1}$ gauss. This is certainly detectable by modern technology. Hence, we conclude that if the electric field is oscillating sufficiently rapidly then electric induction of magnetic fields is an observable effect. In fact, there is a virtually infallible rule for deciding whether or not the displacement current can be neglected in Equation (2.479). Namely, if electromagnetic radiation is important then the displacement current must be included. On the other hand, if electromagnetic radiation is unimportant then the displacement current can be safely neglected. Clearly, Maxwell's inclusion of the displacement current in Equation (2.479) was a vital step in his later realization that his equations allowed propagating wave-like solutions. These solutions are, of course, electromagnetic waves.