Question

(a) Obtain an expression for the mutual inductance between a long straight wire and a square loop of side an as shown in the figure.

(b) Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity, v = 10 m/s.

Calculate the induced emf in the loop at the instant when x = 0.2 m.

Take a = 0.1 m and assume that the loop has a large resistance.

Answer 1

(a) Take a small element dy in the loop at a distance y from the long straight wire (as shown in the given figure).

Magnetic flux associated with elementdy, ${\displaystyle \text{d}\varphi }$ = BdA

Where,

dA = Area of element dy = a dy

B = Magnetic field at distance y

= ${\displaystyle \frac{{\mu }_0\text{I}}{2\pi \text{y}}}$

I = Current in the wire

${\displaystyle {\mu }_0}$ = Permeability of free space = 4π × 10⁻⁷ T m A⁻¹

∴ ${\displaystyle \text{d}\varphi =\frac{{\mu }_0\text{Ia}}{2\pi }\frac{\text{dy}}{\text{y}}}$

${\displaystyle \varphi =\frac{{\mu }_0\text{Ia}}{2\pi }\int \frac{\text{dy}}{\text{y}}}$

y tends from x to a + x

∴ ${\displaystyle \varphi =\frac{{\mu }_0\text{Ia}}{2\pi }{\int }_{\text{x}}^{\text{a}+\text{x}}\frac{\text{dy}}{\text{y}}}$

= ${\displaystyle \frac{{\mu }_0\text{Ia}}{2\pi }{\left[{\log }_{\text{e}}\text{y}\right]}_{\text{x}}^{\text{a}+\text{x}}}$

= ${\displaystyle \frac{{\mu }_0\text{Ia}}{2\pi }{\log }_{\text{e}}\left(\frac{\text{a}+\text{x}}{\text{x}}\right)}$

for mutual inductunce M, the flux is given as :

${\displaystyle \varphi =\text{MI}}$

∴ MI = ${\displaystyle \frac{{\mu }_0\text{Ia}}{2\pi }{\log }_{\text{e}}(\frac{\text{a}}{\text{x}}+1)}$

M = ${\displaystyle \frac{{\mu }_0\text{a}}{2\pi }{\log }_{\text{e}}(\frac{\text{a}}{\text{x}}+1)}$

(b) Emf induced in the loop, e = B’av = ${\displaystyle \left(\frac{{\mu }_0\text{I}}{2\pi \text{x}}\right)\text{av}}$

Given,

I = 50 A

x = 0.2 m

a = 0.1 m

v = 10 m/s

${\displaystyle \text{e}=\frac{4\pi \times {10}^{-7}\times 50\times 0.1\times 10}{2\pi \times 0.2}}$

${\displaystyle \text{e}=5\times {10}^{-5}}$ V