16.1 Maxwell’s equations and electromagnetic waves  (Page 2/8)

and is independent of the surface S through which the current I is measured.

We can now examine this modified version of Ampère’s law to confirm that it holds independent of whether the surface $S_1$ or the surface $S_2$ in [link] is chosen. The electric field $\overrightarrow{E}$ corresponding to the flux ${\text{Φ}}_E$ in [link] is between the capacitor plates. Therefore, the $\overrightarrow{E}$ field and the displacement current through the surface $S_1$ are both zero, and [link] takes the form

We must now show that for surface $S_2,$ through which no actual current flows, the displacement current leads to the same value ${\mu }_{0}I$ for the right side of the Ampère’s law equation. For surface $S_2,$ the equation becomes

Gauss’s law for electric charge requires a closed surface and cannot ordinarily be applied to a surface like $S_1$ alone or $S_2$ alone. But the two surfaces $S_1$ and $S_2$ form a closed surface in [link] and can be used in Gauss’s law. Because the electric field is zero on $S_1$ , the flux contribution through $S_1$ is zero. This gives us

Therefore, we can replace the integral over $S_2$ in [link] with the closed Gaussian surface $S_1+S_2$ and apply Gauss’s law to obtain

Thus, the modified Ampère’s law equation is the same using surface $S_2,$ where the right-hand side results from the displacement current, as it is for the surface $S_1,$ where the contribution comes from the actual flow of electric charge.

Maxwell’s equations

With the correction for the displacement current, Maxwell’s equations take the form

Once the fields have been calculated using these four equations, the Lorentz force equation