Question

The current density across a cylindrical conductor of radius $R$ varies in magnitude according to the equation $J=J_0(1-\frac{r}{R})$ where $r$ is the distance from the central axis. Thus, the current density is a maximum $J_0$ at that axis $(r=0)$ and decreases linearly to zero at the surface $(r=R$). Find the value of current in terms of $J_0$ and conductor's cross-sectional area $A$.

Answer 1

Given: the current density across a cylindrical conductor of radius $R$ varies in magnitude according to the equation $J=J_0(1-\frac{r}{R})$ where $r$ is the distance from the central axis. Thus, the current density is a maximum $J_0$ at that axis $(r=0)$ and decreases linearly to zero at the surface $(r=R)$

To find the value of current in terms of $J_0$ and conductor's cross-sectional area $A$.

Solution:

$J=J_0(1-\frac{r}{R})$

$I={\int }_0^{R}JdA$

$dA=2\pi rdr$, considering it is a cylinder, so consider a ring element and hence the differential area would come out to be this

On integrating we get

$I=J_0\times \pi \times \frac{R^2}{3}$