Question

A conducting loop of radius \( \frac{10}{\sqrt{\pi}} \) cm is placed perpendicular to a uniform magnetic field of 0.5T. The magnetic field is decreased to zero in 0.5 s at a steady rate. The induced emf in the circular loop at 0.25s is:

• A. emf = 1 mV
• B. emf = 10 mV
• C. emf = 100 mV
• D. emf = 5 mV

Hint:

The induced emf is directly proportional to the rate of change of the magnetic flux through the loop. The flux depends on both the magnetic field and the area of the loop.

Answer 1

The Correct Option is B

As \( \varepsilon \big|_{t=0.5 \, \text{sec}} = -\frac{d\phi}{dt} \) \[ = - A \frac{dB}{dt} \quad [\because \theta = 0^\circ \Rightarrow \cos \theta = 1] \] \[ = - \pi \times \left( \frac{10}{\sqrt{\pi}} \right)^2 \times 10^{-4} \times \frac{0 - 0.5}{0.5} = 10^{-2}V = 10 \, \text{mV} \] As \( \frac{dB}{dt} = \) constant \(\Rightarrow\) Induced emf will not change with time. So, \[ \varepsilon \big|_{0.5 \, \text{sec}} = \varepsilon \big|_{0.25 \, \text{sec}} = 10 \, \text{mV} \]