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GATE 2022 EE – Question 23

Electromagnetic Fields · Biot-Savart law, Ampere law, Curl, Faraday law, Lorentz force · 1 mark · Multiple choice

A long conducting cylinder having a radius 'b' is placed along the z axis. The current density is $\mathbf J=J_ar^3\,\hat z$ for the region $r<b$ where $r$ is the distance in the radial direction. The magnetic field intensity ($\mathbf H$) for the region inside the conductor (i.e. for $r<b$) is

  1. $\dfrac{J_a}{4}r^4$
  2. $\dfrac{J_a}{3}r^3$
  3. $\dfrac{J_a}{5}r^4$
  4. $J_ar^3$

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Correct answer: (C) $\dfrac{J_a}{5}r^4$

Explanation

By Ampere's law, $H\cdot2\pi r=\int_0^rJ_ar'^3\,2\pi r'\,dr'=2\pi J_a\frac{r^5}{5}$. So $H=\frac{J_a}{5}r^4$ (in the $\hat\phi$ direction).