Question:

A coil C of N = 117 turns is placed around a long solenoid S. The solenoid has a uniform turn density n (turns per meter). The coil is designed to capture nearly all of the magnetic flux generated by the solenoid.

If the current I in the solenoid is time-dependent, what is the mutual inductance M between the solenoid and the coil? Also, determine the expression for the induced emf in the coil using Faraday’s law.

Answer with Detailed Explanation:

Step 1: Magnetic Field of the Solenoid

For a long solenoid with turn density n, the magnetic field inside is:
B = μ₀nI
where:

• μ₀ is the permeability of free space (μ₀ = 4π × 10⁻⁷ H/m)
• I is the solenoid current

Step 2: Magnetic Flux Through One Turn of Coil C

If the solenoid and the coil have the same cross-sectional area A, then the magnetic flux through one turn of the coil is:
Φ = BA = μ₀nIA

Step 3: Total Flux Linkage of the Coil

For N = 117 turns:
Λ = NΦ = Nμ₀nIA

Step 4: Induced emf Using Faraday’s Law

By Faraday’s law:
ε = −dΛ/dt = −Nμ₀nA (dI/dt)

Step 5: Define Mutual Inductance

We define mutual inductance M such that:
Λ = MI
Comparing:
M = Nμ₀nA

Final Expression for emf

The induced emf becomes:
ε = −M (dI/dt)

Final Answer Summary:

• Mutual Inductance: M = Nμ₀nA
• Induced emf in the coil: ε = −M (dI/dt)