The magnetic moments associated with two closely wound circular coils A and B of radius $r_A=10…

The magnetic moments associated with two closely wound circular coils A and B of radius $r_A=10 \mathrm{~cm}$ and $r_B=20 \mathrm{~cm}$ respectively are equal if $\left(\mathrm{N}_A, \mathrm{I}_A\right.$ and $\mathrm{N}_{\mathrm{B}}, \mathrm{I}_{\mathrm{B}}$ are number of turns and current of A and B respectively)
  1. $2 \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$
  2. $\quad \mathrm{N}_{\mathrm{A}}=2 \mathrm{~N}_{\mathrm{B}}$
  3. $\quad N_A \mathrm{I}_A=4 N_B I_B$
  4. $4 \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$

Solution

$\begin{array}{ll} & \text { We know that, } \mathrm{m}=\text { NIA } \\ & \text { Given that, } \mathrm{m}_{\mathrm{A}}=\mathrm{m}_{\mathrm{B}} \\ \therefore \quad & \mathrm{N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}} \mathrm{A}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}} \mathrm{A}_{\mathrm{B}} \\ \therefore \quad & \mathrm{N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}\left[\pi\left(\mathrm{R}_{\mathrm{A}}\right)^2\right]=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}\left[\pi\left(\mathrm{R}_{\mathrm{B}}\right)^2\right] \\ & \mathrm{N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}(0.1)^2=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}(0.2)^2 \\ \therefore \quad & \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=4 \mathrm{~N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}\end{array}$

Asked in: MHT CET 2024 (04 May Shift 1)

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