Question

A long straight wire carrying a current of 0.1 A is placed at a distance of 10 m from a small conducting square loop in the same plane. The side of the square is 11 mm and the resistance of the square is 3 ohms. An external agent gradually changes the shape of the square loop to nearly a circle in 2 seconds. What is the average induced current in the loop?

Answer 1

 

Given: 

Distance between wire and the centre of the loop, d = 10 m

Current in the wire, I = 0.1 A 

Side of the square loop, a = 11 mm 

Resistance of the loop, R = 3 ohm 

Time taken for changing from square to circle, Δt = 2 s 

To find: 

Average induced current = I_avg 

= (Average induced EMF)/Resistance = ΔΦ/RΔt .....(i) 

ΔΦ = BΔA = (μ₀ I/2πd) × ΔA 

(B is magnetic field due to the long straight wire)Substituting the known values we get, 

ΔΦ = (μ₀ × 0.1/2π × 10)× ΔA = 2 10^-9 × ΔA …..(ii) 

When the square changes to circle its perimeter will remain the same so we can say 

2πr = 4a 

r = 2a/π 

Now, ΔA = πr² - a² = π(2a/π)² - a² 

= (4a² /π) - a² 

Substituting the value of a 

ΔA = (4 × 11 × 11 × 7/22)-(11 × 11) = 33 mm² = 33 × 10^-6m²

Substituting value of ΔA in (ii) we have

ΔΦ = 2 × 10^-9 × 33 ×10^-6 = 6.6 × 10^-14 T .m² 

Substituting values in equation (i) 

I_avg = 6.6 × 10^-14/(3 × 2) = 1.1 × 10^-14 A