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 $P$, which is at a position of distance $r>R$ from the axis of the wire?

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

Answer 1

The correct option is D $\frac{{\mu }_{0}CR}{r}$

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

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

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

$\therefore$ $B_p\times 2\pi r={\mu }_o\times 2\pi RC$

$⟹B_p=\frac{{\mu }_{o}CR}{r}$ for $r>R$