Ampère's Law

The magnetic circulation $\Gamma_B$ around the periphery of the capacitor in the right panel of figure 17.2 is easily computed by taking the magnitude of $\mbox{\bf B}$ in equation (17.6). The magnitude of the magnetic field on the inside of the capacitor is just $B = ir/(2 \epsilon_0 c^2 S)$, since $r = (y^2 + z^2 )^{1/2}$ in figure 17.2. Thus, at the periphery of the capacitor, $r = R$, and $B = iR/(2 \epsilon_0 c^2 S)$ there. The area of the capacitor plates is $S = \pi R^2$ and $\epsilon_0 c^2 = 1/ \mu_0$, as we discussed previously. Thus, the magnetic field is $B = \mu_0 i/(2 \pi R)$ at the periphery. If the periphery is traversed in the counter-clockwise direction, the magnetic circulation around the capacitor is $\Gamma_B = 2 \pi R B = \mu_0 i$.

Let us now compute the magnetic circulation around a wire carrying a current. The magnetic field a distance $r$ from a straight wire carrying a current $i$ is $B = \mu_0 i/(2 \pi r)$. The magnetic field points in the direction of a circle concentric with the wire. The magnetic circulation around the wire is thus $\Gamma_B = 2 \pi r B = \mu_0 i$.

Notice that the same magnetic circulation is found to be the same around the wire and around the periphery of the capacitor. Furthermore, this circulation depends only on the current in the wire and the constant $\mu_0$.

One further item needs to be calculated, namely the electric flux across the gap between the capacitor plates. This is just the electric field $E = \sigma / \epsilon_0$ times the area $S$, or $\Phi_E = S \sigma / \epsilon_0 = q/ \epsilon_0$. The current into the capacitor is the time rate of change on the capacitor, so $i = dq/dt = \epsilon_0 d \Phi_E /dt$.

We are now in a position to understand Ampère's law:

$$\Gamma_B = \mu_0 \left( i + \epsilon_0 \frac{d \Phi_E}{dt} \right) ~~~~ \mbox{(Amp\\lq {e}re's law)} .$$ (18.8)

This states that the magnetic circulation around a loop equals the sum of two contributions, (1) $\mu_0$ times the electric current through the loop and (2) $\mu_0 \epsilon_0$ times the time rate of change of the electric flux through the loop. In the above example the first term dominates when the loop is around the wire, while the second term acts when the loop is around the gap between the capacitor plates.

Ampère actually formulated an incomplete version of the law named after him -- he included only the first term containing the current. The Scottish physicist James Clerk Maxwell added the second term, based primarily on theoretical reasoning. Maxwell's additional term solved a serious internal inconsistency in electromagnetic theory -- in our terms, the Lorentz condition requires a magnetic field to exist if the scalar potential $\phi$ is time-dependent. This magnetic field is only predicted by Ampère's law if Maxwell's term is included. The quantity $\epsilon_0 d \Phi_E /dt$ was called the displacement current by Maxwell since it has the dimensions of current and is numerically equal to the current entering the capacitor. However, it isn't really a current -- it is just the time-changing electric flux!

Gauss's law for electricity and magnetism, Faraday's law, and Ampère's law are collectively called Maxwell's equations. Together they form the basis for electromagnetism as it developed historically. However, our formulation of electromagnetism in terms of the four-potential, the dispersion relation for free electromagnetic waves, the Lorentz condition, and Coulomb's law, is precisely equivalent to Maxwell's equations, and is much closer to the modern approach to electromagnetism.