Inductance of a Solenoid. (b) A metallic laboratory spring is typically 5.00 cm long and 0.150 cm in diameter and has 50 coils. If you connect such a spring in an electric circuit, how much self-inductance must you include for it if you model it as an ideal solenoid?

Verified step by step guidance

1. Calculate the cross-sectional area (A) of the solenoid using the formula for the area of a circle, A = \pi r^2, where r is the radius of the solenoid.
2. Determine the length (l) of the solenoid. Since the spring is coiled, the length of the solenoid is the product of the number of coils and the diameter of each coil.
3. Use the formula for the inductance (L) of a solenoid, L = \mu_0 \frac{N^2}{l} A, where \mu_0 is the permeability of free space (4\pi \times 10^{-7} T\cdot m/A), N is the number of turns, and A and l are the cross-sectional area and length of the solenoid respectively.
4. Substitute the values for N, A, and l into the formula to calculate the inductance.
5. Consider any assumptions made in the calculation, such as the solenoid being tightly wound and air being the core material (ignoring the effects of the metal in the spring).

Key Concepts

Self-Inductance

Self-inductance is a property of a coil or solenoid that quantifies its ability to induce an electromotive force (EMF) in itself due to a change in current. It is measured in henries (H) and depends on the coil's geometry and the material within it. The self-inductance increases with the number of turns in the coil and the cross-sectional area, while it decreases with the length of the coil.

Solenoid

A solenoid is a long coil of wire wound in a helical shape, which generates a uniform magnetic field when an electric current passes through it. The magnetic field inside an ideal solenoid is strong and uniform, making it a useful model for understanding inductance. The inductance of a solenoid can be calculated using its physical dimensions, number of turns, and the permeability of the core material.

Inductance Formula

The inductance (L) of a solenoid can be calculated using the formula L = (μ₀ * N² * A) / l, where μ₀ is the permeability of free space, N is the number of turns, A is the cross-sectional area, and l is the length of the solenoid. This formula highlights how inductance is influenced by the coil's dimensions and the magnetic properties of the material inside it, allowing for the determination of the self-inductance in practical applications.