Mutual Inductance - UNSOLVED PRACTICE SET

Q7. Define mutual inductance. Write its SI unit and dimensional formula.

Q8. Two coils have mutual inductance M = 0.8 H. If the current in the primary coil changes at 5 A/s, calculate the induced EMF in the secondary coil.

Q9. Explain why the coefficient of coupling k is always between 0 and 1.

Q10. Two coaxial solenoids have lengths l₁ and l₂, turns N₁ and N₂, and cross-sectional area A. Derive the expression for their mutual inductance.tion here...

Q11. A transformer has primary inductance L₁ = 2 H and secondary inductance L₂ = 8 H. If the mutual inductance is 3 H, calculate the coefficient of coupling. Is this a good transformer design

Q12. Why is mutual inductance maximum when two coils are wound on the same iron core? Explain.

Q13. Derive the expression for mutual inductance between two long coaxial solenoids. Show that M = μ₀N₁N₂A/l, where N₁ and N₂ are the number of turns, A is the cross-sectional area, and l is the length. Explain the assumptions made.

Q14. Explain the concept of coefficient of coupling between two coils. Derive the relation M = k√(L₁L₂). What does k = 0 and k = 1 signify physically? Give examples of both cases.

Q15. Two coaxial solenoids have the following specifications: Solenoid 1 (primary): length 60 cm, radius 2 cm, 1500 turns. Solenoid 2 (secondary): same length and radius, 800 turns. Both are air-core.

(a) Calculate the self-inductance of each solenoid.

(b) Calculate the mutual inductance between them.

(c) Calculate the coefficient of coupling.

(d) If the current in the primary changes from 0 to 3 A in 0.1 s, calculate the induced EMF in the secondary.

(e) If an iron core with μ_r = 500 is inserted, recalculate all values.

[Given: μ₀ = 4π × 10⁻⁷ T m/A]

Q16. In a school physics lab, a student builds a simple transformer using two coils wound on a cardboard tube. Coil 1 (primary) has N₁ = 400 turns, length 30 cm, radius 2.5 cm. Coil 2 (secondary) has N₂ = 800 turns, same dimensions.

(a) Calculate the mutual inductance between the coils.

(b) The primary is connected to an AC source with V₁ = 12 V, 50 Hz. Calculate the maximum rate of change of current in the primary.

(c) Calculate the induced EMF in the secondary.

(d) If the secondary is connected to a resistor of 100 Ω, calculate the current and power in the secondary.

(e) The student inserts an iron rod through the tube. Explain how this affects the mutual inductance and the secondary voltage. Calculate the new values if μ_r = 600.

[Given: μ₀ = 4π × 10⁻⁷ T m/A]

Q17. A wireless charging pad for a phone uses two coaxial circular coils. The transmitter coil (primary) has N₁ = 20 turns, radius r₁ = 3 cm, and carries current I₁ = 2 sin(2π × 10⁵t) A. The receiver coil (secondary) has N₂ = 30 turns, radius r₂ = 2.5 cm, and is placed 1 cm above the primary.

(a) Calculate the mutual inductance between the coils (approximate using M ≈ μ₀N₁N₂πr₂²/2r₁ for r₂ < r₁).

(b) Calculate the induced EMF in the receiver coil.

(c) If the receiver coil has resistance 2 Ω and is connected to a charging circuit, calculate the power transferred.

(d) Explain why wireless charging is less efficient than wired charging and how the efficiency can be improved.

(e) A student asks why the coils need to be aligned coaxially. What happens if the receiver coil is tilted by 45°? Calculate the approximate change in mutual inductance.

[Given: μ₀ = 4π × 10⁻⁷ T m/A]

Q18. Two circular loops of radii R₁ = 10 cm and R₂ = 5 cm are placed coaxially with their centers 20 cm apart. Loop 1 carries current I₁ = 5 A.

(a) Calculate the magnetic field at the center of Loop 2 due to Loop 1.

(b) Calculate the flux through Loop 2 due to this field (approximate by assuming uniform field over Loop 2).

(c) Calculate the mutual inductance M₁₂.

(d) By reciprocity, M₂₁ = M₁₂. Verify this by calculating the flux through Loop 1 due to current in Loop 2.

(e) If Loop 2 has 50 turns and Loop 1 current changes at 100 A/s, calculate the induced EMF in Loop 2.

(f) Explain why mutual inductance decreases rapidly with distance between coils.

[Given: μ₀ = 4π × 10⁻⁷ T m/A]