Question

$3$ infinitely long thin wires each carrying current $i$ in the same direction, are in the $x-y$ plane in a gravity-free space. The central wire is along the $y$ -axis while the other two are along with $x=+-d. \, $ If the central wire is displaced along the $z$ -direction by a small amount & released, the wire executes the simple harmonic motion. If the linear density of the wire is $\lambda $ , find the frequency of oscillation.

• A. $\frac{i}{2 \pi d}\sqrt{\frac{\mu _{0}}{\pi \lambda }}$
• B. $\frac{i}{2 \pi d}\sqrt{\frac{\pi \lambda }{\mu _{0}}}$
• C. $\frac{2 \mu d}{i}\sqrt{\frac{\pi \lambda }{\mu _{0}}}$
• D. $\frac{2 \mu d}{i}\sqrt{\frac{\mu _{0}}{\mu \lambda }}$

Answer 1

Answer: (A)

When wire (2) is displaced by x along y-axis force is taken per unit length
$\overset{ \rightarrow }{\textit{F}}_{2}=\overset{ \rightarrow }{\textit{F}}_{1 2}+\overset{ \rightarrow }{\textit{F}}_{2 3}$
$\text{since}\text{F}_{1 2}=F_{2 3}=\frac{\mu _{0}}{4 \pi }\frac{2 i^{2}}{\sqrt{d^{2} + x^{2}}}$
$F_{2}=2F_{1 2}\text{sin}\theta $
$F=2 \frac{\mu_{0}}{4 \pi} \frac{2 i^{2}}{\sqrt{d^{2}+x^{2}}} \times \frac{x}{\sqrt{d^{2}+x^{2}}}=\frac{\mu_{0}}{\pi} \frac{i^{2} x}{\left(d^{2}+x^{2}\right)}$
$\textit{x} < < d$
$\textit{F}_{\text{R}}=-\frac{\mu _{0}}{\pi }\frac{i^{2} x}{d^{2}}\text{comparing with }\textit{F}_{\text{R}}=-\textit{Kx}\Rightarrow \textit{K}=\frac{\mu _{0}}{\pi }\frac{i^{2} x^{2}}{d^{2}}$
$\textit{n}=\frac{1}{2 \pi }\sqrt{\frac{K}{m}}=\frac{1}{2 \pi }\sqrt{\frac{\mu _{0} i^{2}}{\pi d^{2} \times \lambda }}=\frac{i}{2 \pi d}\sqrt{\frac{\mu _{0}}{\pi \lambda }}$