Current Density — Question 10

Question 10

A conducting sheet has current per width rather than per area

A very thin strip occupies $0<y<w$ and carries sheet current density $\vec K(y)=K_0(1-y/w)\hat x$, measured in A/m. Its thickness is h, with current uniform through the thickness.

1. Find the total current and the equivalent volume current density.

2. Find the width measured from y=0 that carries 75 percent of the current.

Question 10 - Solution

A conducting sheet has current per width rather than per area

(a) Method and calculation

$$I=\int_0^w K(y)\,dy=K_0w/2.$$$$\vec J(y)=\vec K(y)/h=\frac{K_0}{h}(1-y/w)\hat x.$$

(b) Interpretation and further calculation

$$I(y)/I=2u-u^2,\quad u=y/w.$$$$2u-u^2=.75\Rightarrow u=.5\Rightarrow y=w/2.$$

Physical check

The units of K supply the missing thickness integral; treating K as A/m squared would be dimensionally wrong.