A 0.100-m long solenoid has a radius of 0.050 m and 1.50 × 10⁴ turns.The current in the solenoid changes at a rate of 6.0 A/s.A conducting loop of radius 0.0200 m is placed at the center of the solenoid with its axis the same as that of the solenoid as shown.

-Determine the induced emf in the loop.

Question

A 0.100-m long solenoid has a radius of 0.050 m and 1.50 × 10⁴ turns.The current in the solenoid changes at a rate of 6.0 A/s.A conducting loop of radius 0.0200 m is placed at the center of the solenoid with its axis the same as that of the solenoid as shown.
 
-Determine the induced emf in the loop.

Quizplus · Accepted Answer

The induced emf in the loop can be found using the equation:
emf = -N(dΦ/dt)
where N is the number of turns in the loop and dΦ/dt is the rate of change of magnetic flux through the loop. 
The magnetic flux through the loop can be found using the equation:
Φ = BA
where B is the magnetic field at the center of the solenoid and A is the area of the loop. 
The magnetic field at the center of the solenoid can be found using the equation:
B = μ₀nI
where μ₀ is the permeability of free space, n is the number of turns per unit length in the solenoid, and I is the current in the solenoid.

Plugging in the given values and solving for emf, we get: 
B = (4π×10⁻⁷ T·m/A)(1.50×10⁴ turns/π(0.050 m)²)(6.0 A/s) = 0.45 T
A = π(0.0200 m)² = 1.26×10⁻³ m²
dΦ/dt = d(BA)/dt = A(dB/dt) = A(μ₀n)(dI/dt) = (4π×10⁻⁷ T·m/A)(1.50×10⁴ turns/π(0.050 m)²)(6.0 A/s) = 0.45 T·m²/s
emf = -(100 turns)(0.45 T·m²/s) = -45 V
Taking the absolute value, the induced emf in the loop is 45 V, which is equivalent to 1.4×10² V, or choice B.

A 0100-M Long Solenoid Has a Radius of 0