Q4. Assertion: inserting a soft iron core into a solenoid greatly increases its self-inductance. Reason: L = μrμ0n2Al, and soft iron has a very large relative permeability. Are both true, and does the reason explain the assertion?
Both are true, and the reason explains the assertion: L = μrμ0n2Al, and soft iron’s μr is very large (NCERT Eq. (6.16), page 168). Beyond the NCERT text: this holds for a long, uniformly filled solenoid while the core is below saturation, where L is approximately constant (Wikipedia, “Inductance”).
Q5. Two long coaxial solenoids are 0.40 m long. The inner one has radius 2.0 cm and 500 turns per metre; the outer has 800 turns per metre. Find their mutual inductance, and the emf in the inner solenoid when the outer one’s current changes at 25 A/s.
M = μ0n1n2πr12l = 4π × 10−7 × 500 × 800 × π × (0.020)2 × 0.40 ≈ 2.53 × 10−4 H, so |ε| = M dI/dt ≈ 2.53 × 10−4 × 25 ≈ 6.3 × 10−3 V (NCERT Eq. (6.9), page 166).
Q6. The current in a 2.0 H inductor rises from 1.0 A to 3.0 A. How much extra energy is stored?
½L(I22 − I12) = ½ × 2.0 × (9.0 − 1.0) = 8.0 J (NCERT Eq. (6.17), page 169).
Q7. Find the magnetic energy density in a field of 0.50 T in air.
uB = B2/(2μ0) = 0.25/(2 × 4π × 10−7) ≈ 9.9 × 104 J m−3 (NCERT Eq. (6.18), page 170).
Q8. Why is it hard to switch off a large current in a coil quickly, and why may a spark jump across the switch?
A fast fall in current means a large dI/dt, so the back emf L dI/dt is large; it acts along the current to keep it going, and the high voltage across the opening switch can make a spark (reasoned from NCERT Eq. (6.14), page 168).