Question

The outside of an ideal solenoid $\left(N_{1} \text { turns, length } L\right.\( radius \)r)\( is wound with a coil of wire with \)N_{2}$ turns. (a) What is the mutual inductance? (b) If the current in the solenoid is changing at a rate \(\Delta I_{1} / \Delta t,\) what is the magnitude of the induced emf in the coil?

Short answer

Question: Calculate (a) the mutual inductance between a solenoid of length L, radius r, and N1 turns, and a wire coil wrapped around the solenoid having N2 turns, and (b) the induced emf in the wire coil when the current in the solenoid changes at a rate of ΔI1/Δt. Answer: (a) The mutual inductance, M, between the solenoid and the wire coil can be calculated using the formula: \[M = \frac{N_2 \Phi}{I_1}\] (b) The induced emf, |𝜖|, in the wire coil can be calculated using Faraday's law: \[|\varepsilon| = |M \frac{\Delta I_1}{\Delta t} |\] To find the mutual inductance (M) and induced emf (|𝜖|), first calculate the magnetic field inside the solenoid (B) using the formula B = μ₀nI₁, then calculate the magnetic flux linked with the wire coil (Φ) using Φ = B ⋅ A. Insert these values into the formulas for M and |𝜖|, respectively.

Step-by-step solution

Calculate the magnetic field inside the solenoid

To calculate the mutual inductance, we should first find the magnetic field inside the solenoid. Magnetic field in a solenoid can be represented by the following formula: \[B = \mu_0 n I_1\] Here, B is the magnetic field, μ₀ is the permeability of free space (μ₀ = 4π × 10⁻⁷ Tm/A), n is the number of turns per unit length (n = N1/L), and I1 is the current flowing through N1 turns inside the solenoid.

Calculate the magnetic flux linked with the wire coil

With the magnetic field inside the solenoid, we can now calculate the magnetic flux linked with the wire coil wrapped around the solenoid. The magnetic flux is given by: \[\Phi = B \cdot A\] Here, Φ is the magnetic flux, B is the magnetic field calculated in step 1, and A is the cross-sectional area of the solenoid. Since the solenoid is cylindrical in shape, its cross-sectional area is given by 𝜋r².

Calculate the mutual inductance

Now, we can calculate the mutual inductance between the solenoid and the wire coil. Mutual inductance M is given by: \[M = \frac{N_2 \Phi}{I_1}\] Here, N2 is the number of turns of the wire coil, Φ is the magnetic flux calculated in step 2 and I1 is the current flowing through N1 turns inside the solenoid.

Calculate the induced emf in the wire coil

Lastly, we can calculate the induced emf in the wire coil using Faraday's law which states that induced emf is equal to the rate of change of magnetic flux. The formula is as follows: \[|\varepsilon| = |M \frac{\Delta I_1}{\Delta t} |\] Here, |𝜖| is the absolute value of the induced emf, M is the mutual inductance calculated in step 3, and ΔI1/Δt is the rate of change of current in the solenoid. After calculating the induced emf in the wire coil using the provided formula and values, we will obtain the desired results for parts (a) and (b) of the given exercise.