Question

Suppose that the radius of cross-section of the wire used in the previous problem is r. Find the increase in the radius of the loop if the magnetic field is switched off. Young's modulus of the material of the wire is Y.

Answer 1

Given:
Radius of cross-section of the wire = r
Young's modulus of the material of the wire is Y.
As per the question, when the applied magnetic field is switched off, the tension in the wire increased and so did its length.
Young's modulus,
$$\begin{gathered} Y=\frac{\mathrm{Stress}}{\mathrm{Strain}} \\ Y=\frac{T/A}{∆l/l} \end{gathered}$$
Here,
T is the tension
A is the area of cross-section
Δl is the increse in length of the wire
$$\begin{gathered} ∆l=\frac{Tl}{YA}=\frac{iaB\mathit{\times }l}{Y\times \mathrm{\pi }{r}^2} \\ 2\mathrm{\pi }\times ∆r=\frac{iaB\times 2\mathrm{\pi }a}{Y\times \mathrm{\pi }{r}^2} \\ ∆r=\frac{i{a}^{2}B}{\mathrm{\pi }{r}^{2}Y} \end{gathered}$$