question_answer

The current density \[\vec{j}\] inside a long, solid, cylindrical wire of radius a=12 mm is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to\[J=\frac{{{J}_{0}}r}{a}\], where \[{{J}_{0}}=\frac{{{10}^{5}}}{4\pi }A/{{m}^{2}}.\] Find the magnitude of the magnetic field at in\[\mu T\]

A) \[10\mu T\]

B) \[4\mu T\]

C) \[5\mu T\]

D) \[3\mu T\]

Correct Answer: A

Solution :

[a] Current in the element \[=J(2\pi r.dr)\] Current enclosed by ampere loop of radius \[a/2\] \[I=\int\limits_{0}^{a/2}{\frac{{{J}_{0}}r}{a}.2\pi }r.dr\] \[=\frac{2\pi {{J}_{0}}}{3a}{{\left( \frac{a}{2} \right)}^{3}}=\frac{\pi {{J}_{0}}{{a}^{3}}}{12}\] Applying ampere's law \[B.2\pi .\frac{a}{2}={{\mu }_{0}}.\frac{\pi {{J}_{0}}{{a}^{2}}}{12}\Rightarrow B=\frac{{{\mu }_{0}}{{J}_{0}}a}{12}\] On putting values, \[B=10\mu T\]