Calculating the Field from the Potential — Question 10

Question 10

A vacuum sinusoidal boundary pattern decays away from its surface

For z>0 a potential is $V(x,z)=V_0e^{-\kappa z}\cos(\kappa x)$, independent of y, with $V_0,\kappa>0$.

1. Find E and show that its magnitude at a fixed height is independent of x.

2. Verify that the region is charge-free using the Laplacian and find the height at which the field magnitude halves.

Question 10 - Solution

A vacuum sinusoidal boundary pattern decays away from its surface

(a) Method and calculation

$$E_x=\kappa V_0e^{-\kappa z}\sin(\kappa x),\quad E_z=\kappa V_0e^{-\kappa z}\cos(\kappa x),\quad E_y=0.$$$$|\mathbf E|=\kappa V_0e^{-\kappa z}.$$

(b) Interpretation and further calculation

$$\nabla^2V=(-\kappa^2+\kappa^2)V=0\Rightarrow\rho=-\epsilon_0\nabla^2V=0.$$$$e^{-\kappa z_{1/2}}=1/2\Rightarrow z_{1/2}=\ln2/\kappa.$$

Physical check

The field direction varies across each horizontal line even though its magnitude there is constant.