Question

The figure shows the cross-section of a long conducting cylinder of inner radius a and outer radius b The cylinder carries a current whose current density $J=C r^{2}$ where C is a constant. What is the magnitude of the magnetic field $B$ at a point $r$, where $a<\,r<\,b$ ?

• A. $B=\frac{\mu_{0} \pi C}{4 r^{2}}\left(r^{2}-a^{2}\right)$
• B. $B=\frac{\mu_{0} C}{4 r^{2}}\left(r^{4}-a^{4}\right)$
• C. $B=\frac{\mu_{0} \pi C}{4 r}\left(r^{2}-a^{2}\right)$
• D. $B=\frac{\mu_{0} C}{4 r}\left(r^{4}-a^{4}\right)$

Answer 1

Answer: (D)

Given: Current density $J=C r^{2}$
For a region $a<\,r<\,b$. The Amperian loop is a circle labelled as 1 According to Ampere's circuital law,
$\oint \vec{B} \cdot d \vec{l}=\mu_{0} I$
$B(2 \pi r)=\mu_{0} \int J d A=\mu_{0} \int\limits_{a}^{r} C r' 2\left(2 \pi r'd r'\right)$
$=\mu_{0} C 2 \pi \int\limits_{a}^{r} r' 3\,d r'=\mu_{0} C 2 \pi \left[\frac{r' 4}{4}\right]_{a}^{r}$
$B 2 \pi r=\frac{\mu_{0} C 2 \pi}{4}\left[r^{4}-a^{4}\right], B=\frac{\mu_{0} C}{4 r}\left[r^{4}-a^{4}\right]$