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.

A 0.100-m long solenoid has a radius of 0.050 m and 1.50 × 10<sup>4</sup> 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. A) 0.7 × 10<sup>-</sup><sup>3</sup> V B) 1.4 × 10<sup>-</sup><sup>3</sup> V C) 2.8 × 10<sup>-</sup><sup>3</sup> V D) 5.6 × 10<sup>-</sup><sup>3</sup> V E) zero volts

-Determine the induced emf in the loop.

Quizplus · Accepted Answer

The magnetic flux through the loop is given by:
Φ = BA = μₒnIπr²
where,
μₒ = magnetic constant
n = number of turns per unit length
I = current in the solenoid
r = radius of the solenoid
B = magnetic field inside the solenoid
πr² = area of the loop
Differentiating both sides with respect to time, we get:
dΦ/dt = μₒnπr²(dI/dt)
The induced emf is given by:
ε = -dΦ/dt
Substituting the given values, we get:
ε = -μₒnπr²(dI/dt)
ε = -(4π×10⁻⁷ T m/A)×(1.50×10¹⁴ m⁻¹)×π(0.050 m)²×6.0 A/s
ε = -1.425 V
Since the loop is symmetrically placed at the center of the solenoid, the induced emf will be zero if the loop has a circular shape. However, since the loop has a finite radius, there will be a non-zero induced emf due to the changing magnetic field. The negative sign in the above equation indicates that the induced emf is in the opposite direction to the rate of change of current in the solenoid. Therefore, the correct answer is choice B, 1.4 × 10⁻⁹ V.

A 0100-M Long Solenoid Has a Radius of 0