Question

A long solenoid (cross-sectional area =1.3×10⁻⁶m², number of turns per unit length =2448 turns /m ) is bent into a circular shape so it looks like a doughnut. This wire-wound doughnut is called a toroid. Assume that the diameter of the solenoid is small compared to the radius of the toroid, which is 0.068 m. Find the emf induced in the toroid when the current decreases from 2.5 A to 1.1 A in a time of 0.15 s.

Answer 1

The induced emf in a toroid can be calculated using Faraday's law of electromagnetic induction, which involves the change in current, the radius of the toroid, cross-sectional area, the number of turns per unit length, and the magnetic permeability of free space.

Step-by-step explanation:

To find the emf induced in the toroid, we can use Faraday's law of electromagnetic induction. The emf (electromotive force) in a coil is equal to the negative rate of change of magnetic flux through the coil, given by ε = -dΦ/dt. The change in magnetic flux (ΔΦ) due to a change in current (I) in a solenoid is given by Φ = (N * A * B), where N is the total number of turns, A is the cross-sectional area, and B is the magnetic field induced by the solenoid. For a solenoid, B = μ0 * (n * I), where μ0 is the permeability of free space and n is the number of turns per unit length.

In a toroidal solenoid, N = n * 2 * π * r, where r is the radius of the toroid. The change in current ΔI = 2.5 A - 1.1 A = 1.4 A over time Δt = 0.15 s. Plugging these values into Faraday's law gives us:

1. Calculate the total number of turns: N = n * 2 * π * r
2. Determine the change in flux: ΔΦ = N * A * ΔB = N * A * μ0 * ΔI
3. Finally, the induced emf: ε = -ΔΦ/Δt = -(N * A * μ0 * ΔI)/Δt

Substitute the given values: area (A) = 1.3×10⁻⁶ m², the number of turns per unit length (n) = 2448 turns/m, radius (r) = 0.068 m, the magnetic permeability of free space (μ0) = 4π×10⁻⁷ H/m.

With all the information provided, we can solve for the induced emf in the toroid.