Two circular coils 1 and 2 are made from the same wire. The radius of the first coil is twice that of the second coil. What is the ratio of potential difference applied across them V₁ / V₂ , so that the magnetic field at their centre is the same?

Step-by-step solution

Magnetic field due to first coil and second coil is same at centre. Then, magnetic field at centre is B= ₀ I₁ 2(2 r) = ₀ I₂ 2(r) I₁ I₂ =2...(i) As we know that, resistance of the coil is related as R= l A where, = resistivity l= length A= area of cross-section and A are same for both coils but l₁=2 (2 r) and l₂=2 (r) . l₂ l₁ = 1 2 ...(ii) If V₁ and V₂ are potential difference applied across first and second coil, then and aligned I₁ & = V₁ R₁ and I₂= V₂ R₂ I₁ & = V₁ l₁ A ...(iii) I₂ & = V₂ l₂ A ...(iv) aligned From