Question

Two concentric, circular wire loops of radii r1 and r2 (r1 < r2) are located in the x-y plane; each carries a current I in a clockwise direction. (a) Find the net dipole moment of this system. (b) Repeat for the reversed current in the inner loop

Answer 1

a. πi(r₁² + r₂²)

b. πi(r₂²- r₁²)

Step-by-step explanation:

(a) Find the net dipole moment of this system.

The magnetic dipole moment, μ = iA where i = current and A = For area

For the circular loop with radius r₁, and current i₁ in a clockwise direction, i₁ = + i positive

μ₁ = i₁A₁

= +i(πr₁²)

= πir₁²

For the circular loop with radius r₂, and current i₂ in a clockwise direction, i₂ = +i

μ₂ = i₂A₂

= +i(πr₂²)

= πir₂²

The net dipole moment, μ = μ₁ + μ₂

μ = μ₁ + μ₂

= πir₁² + πir₂²

= πi(r₁² + r₂²)

(b) Repeat for the reversed current in the inner loop

With the reversed current in the inner loop, the current is negative. So i₂ =-i

μ₁ = i₂A₁ = -i(πr₁²) = -πir₁²

So,

μ = μ₁ + μ₂

= -πir₁² + πir₂²

= πi(-r₁² + r₂²)

= πi(r₂²- r₁²)