question_answer

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) \[B=\frac{{{\mu }_{0}}\pi C}{4{{r}^{2}}}({{r}^{2}}-{{a}^{2}})\]

B) \[B=\frac{{{\mu }_{0}}C}{4{{r}^{2}}}({{r}^{4}}-{{a}^{4}})\]

C) \[B=\frac{{{\mu }_{0}}\pi C}{4r}({{r}^{2}}-{{a}^{2}})\]

D)   \[B=\frac{{{\mu }_{0}}C}{4r}({{r}^{4}}-{{a}^{4}})\]

Correct Answer: D

Solution :

[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 Amperes circuital law \[=\oint{{\vec{B}}}.d\vec{l}={{\mu }_{0}}I\] \[B(2\pi r)={{\mu }_{0}}\int_{{}}^{{}}{JdA}={{\mu }_{0}}\int\limits_{a}^{r}{Cr{{'}^{2}}}(2\pi r'dr')\] \[={{\mu }_{0}}C2\pi \int\limits_{a}^{r}{r{{'}^{3}}dr'}={{\mu }_{0}}C2\pi \left[ \frac{r{{'}^{4}}}{4} \right]_{a}^{r}\] \[B2\pi r=\frac{{{\mu }_{0}}C2\pi }{4}[{{r}^{4}}-{{a}^{4}}];B=\frac{{{\mu }_{0}}C}{4r}[{{r}^{4}}-{{a}^{4}}]\]