A toroidal solenoid has mean radius 12.0 cm and crosssectional area 0.600 cm^2. (a) How many turns does the solenoid have if its inductance is 0.100 mH?

Verified step by step guidance

1. Identify the given values: mean radius of the toroid (R) = 12.0 cm, cross-sectional area (A) = 0.600 cm^2, and inductance (L) = 0.100 mH.
2. Convert the units of the radius and the cross-sectional area to meters: R = 0.12 m, A = 0.000060 m^2.
3. Use the formula for the inductance of a toroidal solenoid: L = \( \mu_0 \frac{N^2 A}{2 \pi R} \), where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \) T·m/A), N is the number of turns, A is the cross-sectional area, and R is the mean radius.
4. Rearrange the formula to solve for N (the number of turns): N = \( \sqrt{\frac{L \cdot 2 \pi R}{\mu_0 A}} \).
5. Substitute the values into the rearranged formula to calculate N.

Key Concepts

Inductance

Inductance is a property of an electrical component that quantifies its ability to store energy in a magnetic field when an electric current flows through it. It is measured in henries (H) and is defined as the ratio of the induced electromotive force (EMF) to the rate of change of current. In the context of a solenoid, inductance depends on factors such as the number of turns, the cross-sectional area, and the core material.

Toroidal Solenoid

A toroidal solenoid is a coil of wire shaped like a doughnut, where the wire is wound around a circular core. This configuration allows for a uniform magnetic field within the core and minimizes magnetic field leakage. The inductance of a toroidal solenoid can be calculated using its geometry, including the mean radius and cross-sectional area, along with the number of turns of wire.

Formula for Inductance of a Toroidal Solenoid

The inductance (L) of a toroidal solenoid can be calculated using the formula L = (μ₀ * N² * A) / (2 * π * r), where μ₀ is the permeability of free space, N is the number of turns, A is the cross-sectional area, and r is the mean radius. This formula highlights the relationship between the solenoid's physical dimensions and its inductance, allowing for the determination of the number of turns when the inductance and other parameters are known.