Question

Two concentric co-planar circular Loops of radii r₁ and r₂ carry currents of respectively I₁ and I₂ in opposite directions. The magnetic induction at the centre of the Loops is half that due to I₁ alone at the centre. If r₂ = 2r₁ the value of (I₂/I₁) is 

(a) 2

(b) 1/2 

(c) 1/4

(d) 1

Answer 1

 The correct option  (d) 1

Explanation:

For the loop, B = (μ₀I/2r)

For smaller loop,   B₁ = (μ₀I₁/2r₁)

For bigger loop,   B₂ = (μ₀I₂/2r₂)

but r₁ < r₂

hence B₁ > B₂

∴   B = B₁ – B₂ = [(μ₀I₁)/(2r₁)] – [(μ₀I₂)/(2r₂)]

r₂ = 2r₁

∴  B = [(μ₀I₁) / (2r₁)] – [(μ₀I₂)/(4r₁)]

∴  B = (μ₀/2r₁) [I₁ – (I₂/2)]

given is B = (1/2) B₁

∴  (1/2)[(μ₀I₁)/(2r₁)] = (μ₀/2r₁) [I₁ – (I₂/2)]

∴  (I₁/2) = I₁ – (I₂/2)

∴  (I₂/2) = (I₁/2)

∴  (I₁/I₂) = 1