Question

A toroid of a circular cross-section of radius 0.05 m has 2000 windings and a self-inductance of 0.04 H. What is (a) the current through the windings when the energy in its magnetic field is 2 × 10⁻⁶ J and (b) the central radius of the toroid?

Answer 1

Data: r = 0.05 m, N = 2000, L = 0.04 H, U_m = 2 × 10⁻⁶ J, μ₀ = 4π × 10⁻⁷ H/m

(a) U_m = `1/2 LI^2`

Therefore, the current in the windings,

I = `sqrt((2U_m)/L)`

= `sqrt((2(2xx 10^-6))/(40 xx 10^-2)`

= 10⁻² A

(b) L = `(μ_0N^2r^2)/(2R)`

Therefore, the central radius of the toroid,

R = `(μ_0N^2r^2)/(2L)`

= `((4π xx 10^-7)(2 xx 10^3)^2 (5 xx 10^-2)^2)/(2(4 xx 10^-2))`

= `π/2 xx 10^-1`

= 0.157 m