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 is the magnetic field at point Q, which is at a position a distance $r<R$ from the axis of the wire?

• A. $\frac{{\mu }_{0}C}{2}$
• B. ${\mu }_{0}C$
• C. $\frac{{\mu }_{0}Cr}{4}$
• D. $\frac{{\mu }_{0}C}{r}$
• E. $\frac{{\mu }_{0}C}{2r}$

Answer 1

The correct option is B ${\mu }_{0}C$

Current enclosed by the loop of radius $r<R$,

$I_{enc}=2\pi rC$ (solved earlier)

Using Ampere's law : $\oint B.dl={\mu }_o{I}_{enc}$

$\therefore$ $B_Q\times 2\pi r={\mu }_o\times 2\pi rC$

$⟹B_Q={\mu }_{o}C$ for $r<R$