Question

A circular loop of wire of radius \(14\,\text{cm}\) is placed in a magnetic field directed perpendicular to the plane of the loop. If the field decreases at a steady rate of \(0.05\,\text{T s}^{-1}\) in some interval, then the magnitude of the emf induced in the loop is

• A. \(2.08\,\text{mV}\)
• B. \(3.08\,\text{mV}\)
• C. \(2.16\,\text{mV}\)
• D. \(3.24\,\text{mV}\)

Hint:

When the magnetic field is perpendicular to the plane of a circular loop, magnetic flux is \[ \Phi=BA \] and induced emf is \[ \varepsilon=A\left|\frac{dB}{dt}\right| \] for a single loop.

Answer 1

The Correct Option is B

Step 1: Recall Faraday's Law of electromagnetic induction.
The magnitude of the emf induced in a conducting loop is equal to the rate of change of magnetic flux through the loop: \[ |\varepsilon| = \left|\frac{d\Phi}{dt}\right| \]
Step 2: Express the magnetic flux.
Since the magnetic field $B$ is perpendicular to the plane of the circular loop, the flux is: \[ \Phi = B \cdot A = B \cdot \pi r^2 \] Therefore: \[ |\varepsilon| = \pi r^2 \left|\frac{dB}{dt}\right| \]
Step 3: Identify the given values.
Radius: $r = 14\,\text{cm} = 0.14\,\text{m}$. Rate of change of field: $\left|\dfrac{dB}{dt}\right| = 0.05\,\text{T s}^{-1}$.
Step 4: Calculate the area of the loop.
\[ A = \pi r^2 = \pi \times (0.14)^2 = \pi \times 0.0196\,\text{m}^2 \]
Step 5: Compute the induced emf.
\[ |\varepsilon| = \pi \times 0.0196 \times 0.05 = \pi \times 9.8 \times 10^{-4} \] \[ |\varepsilon| = 3.14159 \times 9.8 \times 10^{-4} \approx 3.08 \times 10^{-3}\,\text{V} \]
Step 6: State the final answer.
The magnitude of the induced emf in the circular loop is: \[ \boxed{|\varepsilon| \approx 3.08\,\text{mV}} \]