Two concentric coplanar circular loops of radii r1 and r2 carry currents of . . .

Question : Two concentric coplanar circular loops of radii r₁ and r₂ carry currents of respectively i₁ and i₂ in opposite directions (one clockwise and the other anticlockwise.) 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) ½

(c) ¼

(d) 1

Doubt by Diya

Solution : 

We know, 
Magnetic Field at the centre of circular current carrying coil is 
B = μ₀i/2r

So, 
B₁ = μ₀i₁/2r₁ (Downward)

B₂ = μ₀i₂/2r₂ (Upward)

B = B₁-B₂
B₁/2 = B₁-B₂
B₂ = B₁-B₁/2
B₂=B₁/2
2B₂=B₁
2×(μ₀i₂/2r₂) = μ₀i₁/2r₁
2×(i₂/r₂) = i₁/r₁

2×(i₂/2r₁)=i₁/r₁ [∵ r₂ = 2r₁] 

i₂/r₁=i₁/r₁
i₂=i₁
i₂/i₁ = 1

Hence, the correct option is d) 1