Current Density — Question 7

Question 7

A transverse density variation must be integrated across a rectangular contact

A rectangular contact lies in the yz plane, $-a\le y\le a$, $0\le z\le b$, with normal $+\hat x$. Its current density is $\vec J=J_0(1+y/a)^2\hat x+J_t\hat y$.

1. Find the current through the contact.

2. Find the fraction crossing the half with y>0 and explain why $J_t$ is absent.

Question 7 - Solution

A transverse density variation must be integrated across a rectangular contact

(a) Method and calculation

$$I=\int_0^b\int_{-a}^a J_0(1+y/a)^2\,dy\,dz.$$$$I=\frac83J_0ab.$$

(b) Interpretation and further calculation

$$I_{y>0}=J_0ab\int_0^1(1+u)^2du=\frac73J_0ab.$$$$I_{y>0}/I=7/8,\quad(J_t\hat y)\cdot\hat x=0.$$

Physical check

Only the component normal to the chosen surface contributes to its flux.