Question

1
A current is flowing through a cylindrical conductor
of radius R such that current density at radial
distance r is given by J =Jnote
$(1 - r \div r)$
Calculate
total current through cross-section of conductor.
​

Answer 1

you mean, current density at radial distance r is given by, $J_0\left(1-\frac{r}{R}\right)$ , right?

we know, current density is the rate of flowing of current through unit cross sectional area.

i.e., J = dI/dA

or, $I=\int{J}\,dA$

so, first of all, finding elementary area of cylinderical conductor.

A = πr²

differentiating both sides,

dA = 2πr . dr

then, current, I = $\int\limits^R_0{J}\,(2\pi r).dr$

= $2\pi J_0\int\limits^R_0{\left(r-\frac{r^2}{R}\right)}\,dr$

= $2\pi J_0\left[\frac{r^2}{2}-\frac{r^3}{3R}\right]^R_0$

= $2\pi J_0\left[\frac{R^2}{2}-\frac{R^2}{3}\right]$

= $\frac{\pi J_0R^2}{3}$ This is our required answer.