A long solenoid of diameter 0.1 m has 2 × 10⁴ turns per meter. At the centre of the solenoid a coil of 100 turns and radius 0.01 m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to 0 A from 4 A in 0.05 s. If the resistance of the coil is $10 \pi^2 \Omega$, then the total charge flowing through the coil during this time is _________.

Fill in the blank with the correct answer from the options given below.

$32 \mu C$

The correct answer is Option 2: $32 \mu C$

Magnetic field inside solenoid:
B = μ₀ n I
= 4π × 10⁻⁷ × 2 × 10⁴ × 4
= 32π × 10⁻³ T

Area of coil:
A = πr² = π(0.01)² = π × 10⁻⁴

Initial flux (at I = 4 A):
Φᵢ = N B A
= 100 × 32π × 10⁻³ × π × 10⁻⁴
= 32π² × 10⁻⁵ Wb

Final flux (at I = 0):
Φ???? = 0

Change in flux:
ΔΦ = Φᵢ − Φ???? = 32π² × 10⁻⁵

Charge flown:
q = ΔΦ / R
= (32π² × 10⁻⁵) / (10π²)
= 32 × 10⁻⁶ C

q = 32 μC