Two concentric circular coils of 'n' turns each are situated in the same plane. Their radii are ' $a_{1}$ '…

Two concentric circular coils of 'n' turns each are situated in the same plane. Their radii are ' $a_{1}$ ' and ' $a_{2}{ }^{\prime}\left(a_{2}>a_{1}\right)$ and they carry currents 'I $_{1}$ ' and ' $\mathrm{I}_{2}$ ' respectively $\left(\mathrm{I}_{1}>\mathrm{I}_{2}\right)$ in opposite direction. The magnetic field at the centre is
  1. $\frac{\mu_{0} n}{2}\left[\frac{\mathrm{I}_{1} \mathrm{a}_{2}-\mathrm{I}_{2} \mathrm{a}_{1}}{\mathrm{a}_{1} \mathrm{a}_{2}}\right]$
  2. $\frac{\mu_{0} n}{2 \mathrm{a}_{1} \mathrm{a}_{2}}\left[\mathrm{I}_{1}-\mathrm{I}_{2}\right]$
  3. $\frac{\mu_{0} n}{2 \mathrm{I}_{1} \mathrm{I}_{2}}\left[\mathrm{a}_{2}-\mathrm{a}_{1}\right]$
  4. $\frac{\mu_{0} n}{2}\left[\frac{\mathrm{I}_{1} \mathrm{a}_{1}-\mathrm{I}_{2} \mathrm{a}_{2}}{\mathrm{a}_{1} \mathrm{a}_{2}}\right]$

Solution

Magnetic fields due to the two coils are in opposite directions and are given by $\begin{array}{l} \mathrm{B}_{1}=\frac{\mu_{0} \mathrm{nI}_{1}}{2 \mathrm{a}_{1}} \quad \text { and } \quad \mathrm{B}_{2}=\frac{\mu_{0} \mathrm{n}_{2}}{2 \mathrm{a}_{2}} \end{array}$ $\begin{aligned} \therefore \mathrm{B} &=\mathrm{B}_{1}-\mathrm{B}_{1} \\ &=\frac{\mu_{0} \mathrm{n}}{2}\left[\frac{\mathrm{I}_{1}}{\mathrm{a}_{1}}-\frac{\mathrm{I}_{2}}{\mathrm{a}_{2}}\right] \\ &=\frac{\mu_{0} \mathrm{nI}}{2}\left[\frac{\mathrm{I}_{1} \mathrm{a}_{2}-\mathrm{I}_{2} \mathrm{a}_{1}}{\mathrm{a}_{1} \mathrm{a}_{2}}\right] \end{aligned}$

Asked in: MHT CET 2020 (19 Oct Shift 1)

Practice more Magnetic Fields due to Electric Current questions on Aicharya