Question

A cylindrical wire of radius R carries a current directed out of the page as shown below. The current has a cross-sectional area density J that varies with distance r from the wire's axis according to the function $J=Cr$.
According to Ampere's Law, the current creates magnetic field both within the wire and outside the wire.
What current is enclosed in a circular loop that is centered on the wire's axis and has a radius $r>R$?

• A. $C{R}^3$
• B. $\frac{C{R}^4}{4}$
• C. $4C{R}^3$
• D. $2\pi \frac{C{R}^3}{3}$
• E. $2C{R}^2$

Answer 1

The correct option is D $2\pi \frac{C{R}^3}{3}$

Given : $J=Cr$

Current enclosed by the loop of radius $r>R$ is equal to the total current flowing through the wire.

$\therefore$ Current enclosed by the loop of radius $r>R$, $I={\int }_o^{R}Jda$

$\therefore$ $I_{enc}={\int }_o^{R}Cr\left(2\pi r\right)dr=2\pi C{\int }_o^R{r}^{2}dr$

OR $I_{enc}=2\pi C\times \frac{R^3}{3}$ $⟹I_{enc}=2\pi \frac{C{R}^3}{3}$ for $r>R$