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: Use Faraday's law of electromagnetic induction.
The magnitude of induced emf in a loop is given by \[ \varepsilon=\left|\frac{d\Phi}{dt}\right| \] For one circular loop, \[ \Phi=BA \] Since the magnetic field is perpendicular to the plane of the loop, \[ \Phi=BA \] Therefore, \[ \varepsilon=A\left|\frac{dB}{dt}\right| \]

Step 2: Find the area of the circular loop.
Given radius, \[ r=14\,\text{cm}=14\times 10^{-2}\,\text{m}=0.14\,\text{m} \] Area of the loop is \[ A=\pi r^2 \] \[ A=\pi(0.14)^2 \] \[ A=\pi(0.0196) \] Using \[ \pi=\frac{22}{7} \] \[ A=\frac{22}{7}\times 0.0196 \] \[ A=0.0616\,\text{m}^2 \]

Step 3: Use the rate of decrease of magnetic field.
The magnetic field decreases at a steady rate of \[ \left|\frac{dB}{dt}\right|=0.05\,\text{T s}^{-1} \] So, \[ \varepsilon=A\left|\frac{dB}{dt}\right| \] \[ \varepsilon=0.0616\times 0.05 \] \[ \varepsilon=0.00308\,\text{V} \]

Step 4: Convert volt into millivolt.
Since, \[ 1\,\text{V}=1000\,\text{mV} \] we get \[ 0.00308\,\text{V}=3.08\,\text{mV} \]

Step 5: Final conclusion.
Therefore, the magnitude of the emf induced in the loop is \[ \boxed{3.08\,\text{mV}} \]