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
${\overrightarrow{F}}_2={\overrightarrow{F}}_{12}+{\overrightarrow{F}}_{23}$
$\text{since}{\text{F}}_{12}=F_{23}=\frac{{\mu }_0}{4\pi }\frac{2i^2}{\sqrt{d^2+x^2}}$
$F_2=2F_{12}\text{sin}\theta$
$F=2\frac{{\mu }_0}{4\pi }\frac{2i^2}{\sqrt{d^2+x^2}}\times \frac{x}{\sqrt{d^2+x^2}}=\frac{{\mu }_0}{\pi }\frac{i^{2}x}{\left(d^2+x^2\right)}$
$\text{x}<<d$
${\text{F}}_{\text{R}}=-\frac{{\mu }_0}{\pi }\frac{i^{2}x}{d^2}\text{comparing with }{\text{F}}_{\text{R}}=-\text{Kx}\Rightarrow \text{K}=\frac{{\mu }_0}{\pi }\frac{i^2{x}^2}{d^2}$
$\text{n}=\frac{1}{2\pi }\sqrt{\frac{K}{m}}=\frac{1}{2\pi }\sqrt{\frac{{\mu }_0{i}^2}{\pi d^2\times \lambda }}=\frac{i}{2\pi d}\sqrt{\frac{{\mu }_0}{\pi \lambda }}$