MHT CET · Physics · Magnetic Effects of Current
Two similar coils each of radius \(\mathrm{R}\) are lying concentrically with their planes at right angles to each other. The current flowing in them are I and 2I. The resultant magnetic field of induction at the centre will be \(\left(\mu_0=\right.\) Permeability of vacuum)
- A \(\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}\)
- B \(\frac{\mu_0 \mathrm{I}}{\mathrm{R}}\)
- C \(\frac{3 \mu_0 \mathrm{I}}{2 \mathrm{R}}\)
- D \(\frac{\sqrt{5} \mu_0 \mathrm{I}}{2 \mathrm{R}}\)
Answer & Solution
Correct Answer
(D) \(\frac{\sqrt{5} \mu_0 \mathrm{I}}{2 \mathrm{R}}\)
Step-by-step Solution
Detailed explanation
\(
\mathrm{B}_1=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}, \mathrm{B}_2=\frac{\mu_0(2 \mathrm{I})}{2 \mathrm{R}}
\)
Resultant magnetic field, \(B=\sqrt{\mathrm{B}_1+\mathrm{B}_2}\)
\(
\begin{array}{ll}
\therefore & B=\sqrt{\left(\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}\right)^2+\left(\frac{\mu_0(2 \mathrm{I})}{2 \mathrm{R}}\right)^2} \\
\therefore & \mathrm{B}=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}} \sqrt{1^2+2^2} \\
\therefore & \mathrm{B}=\frac{\sqrt{5} \mu_0 \mathrm{I}}{2 \mathrm{R}}
\end{array}
\)
\mathrm{B}_1=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}, \mathrm{B}_2=\frac{\mu_0(2 \mathrm{I})}{2 \mathrm{R}}
\)
Resultant magnetic field, \(B=\sqrt{\mathrm{B}_1+\mathrm{B}_2}\)
\(
\begin{array}{ll}
\therefore & B=\sqrt{\left(\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}\right)^2+\left(\frac{\mu_0(2 \mathrm{I})}{2 \mathrm{R}}\right)^2} \\
\therefore & \mathrm{B}=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}} \sqrt{1^2+2^2} \\
\therefore & \mathrm{B}=\frac{\sqrt{5} \mu_0 \mathrm{I}}{2 \mathrm{R}}
\end{array}
\)
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