Physics Bootcamp

(Requires Calculus).

The four basic laws of electricity and magnetism were discovered experimentally. While studying them James Clerk Maxwell(1831-1879) discovered some logical inconsistencies. The source of inconsistencies turned out to be the incompleteness of Ampere's law. Recall that according to Ampere's law the circulation of magnetic field around a loop C is proportional to the current passing through any surface (S) whose boundary is the loop C itself.

Although, there are infinitely many surfaces that can be attached to any loop, it is important to realize that Ampere's law stated in Eq. (45.1.1) is independent of the choice of surface as illustrated in Figure 45.1.1.

Now, consider the following experiment shown in Figure 45.1.2.

Connect a parallel plate capacitor to an EMF source \(V\text{.}\) When the switch is closed, a time-dependent current develops in the wire. Let us now apply Ampere's law to loop C shown at a time before the capacitor is fully charged such that \(I \ne 0\text{.}\) With the surface \(S_1\text{,}\) you get a non-zero value for the enclosed current \(I\text{,}\) while with surface \(S_2\) you get a zero for the enclosed current since no current passes through it.

Clearly, Ampere's law does not work here. You might have already suspected this because Ampere's law requires a steady current, while the current in this experiment is changing with time and not steady at all.

How can we modify Ampere's law so that it will work in all situations? This question was answered by James Clerk Maxwell. The experiment with the capacitor suggests a solution for this problem. In Figure 45.1.2, you will notice that, while current passes through surface \(S_1\text{,}\) a time-dependent electric field passes through \(S_2\text{.}\) So, it would be tempting to correct Eq. (45.1.1) by adding the flux of E-field through a surface in addition to \(I_{\text{enc}}\text{.}\) But this will not work because when the capacitor is fully charged you get another inconsistency: although there will be no current through \(S_1\text{,}\) there will still be a non-zero flux of E-field through \(S_2\text{.}\) Maxwell suggested that rather than adding the electric flux, you will need the rate of change of electric flux to correct the Ampere's law. He suggested the form of the additional contribution to the real current \(I\) called the displacement current \(I_d\text{.}\)

The displacement current \(I_d\) is supposed to be the contribution from the changing electric flux through the surface attached to the loop for the circulation integral on theleft side of this equation. Specifically, the displacement current is defined to be

where \(\epsilon_0\) is the permittivity of free space, and \(\Phi_E\) the electric flux defined as

Note that since surfaces attached to the loop are not closed the electric flux is not equal to the \(Q_{enc}/\epsilon_0\) as would be expected for the electric flux through a closed surface according to Gauss's law.

Recall the relationship of current \(I\) to the current density vector .

Similarly, we also define a displacement current density vector \(\vec J_d\text{.}\)

Putting Eq. (45.1.5) in Eq. (45.1.3) and using Eq. (45.1.4) we deduce

The modified Ampere's law, Eq. (45.1.2) also explains correctly even the time-dependent situations depicted in Figure 45.1.2 - the surface \(S_1\) has contribution from current and the surface \(S_2\) has equal contribution from the displacement current of the time-varying electric field. We will call the modified Ampere's law as Ampere-Maxwell's law.