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 $J=\frac{C}{r}$.
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. $2\pi Cr$
• B. $\frac{C{r}^2}{2}$
• C. $Cr$
• D. $4Cr$
• E. $2C$

Answer 1

The correct option is A $2\pi Cr$

Given : $J=\frac{C}{r}$

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

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

OR $I_{enc}=2\pi C\times r$ $⟹I_{enc}=2\pi rC$ for $r<R$