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?

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.