A conducting circular loop of radius `10/sqrt(pi)` cm is placed perpendicular to a unifrom magnetic firld of 0.5 T.The magnetic field is decreased to zero in 0.5sec at a steady rate . The induced emf in the circular loop at 0.25 s is:`

Options

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

Answer

To solve the problem, we need to find the induced electromotive force (emf) in a conducting circular loop when the magnetic field is decreased from 0.5 T to 0 T over a time interval of 0.5 seconds. We will calculate the induced emf at 0.25 seconds.

### Step 1: Determine the area of the circular loop
The radius \( r \) of the circular loop is given as \( \frac{10}{\sqrt{\pi}} \) cm. We first convert this to meters:
\[ r = \frac{10}{\sqrt{\pi}} \times 10^{-2} \text{ m} = \frac{10^{-1}}{\sqrt{\pi}} \text{ m} \]
The area \( A \) of the circular loop is given by:
\[ A = \pi r^2 = \pi \left(\frac{10^{-1}}{\sqrt{\pi}}\right)^2 = \pi \cdot \frac{10^{-2}}{\pi} = 10^{-2} \text{ m}^2 \]

### Step 2: Determine the change in magnetic field
The magnetic field \( B \) is decreased from 0.5 T to 0 T over 0.5 seconds. The rate of change of the magnetic field \( \frac{dB}{dt} \) can be calculated as:
\[ \frac{dB}{dt} = \frac{B_f - B_i}{\Delta t} = \frac{0 - 0.5}{0.5} = -1 \text{ T/s} \]

### Step 3: Calculate the induced emf
The induced emf \( \mathcal{E} \) in the loop is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -A \frac{dB}{dt} \]
Substituting the values we have:
\[ \mathcal{E} = -10^{-2} \cdot (-1) = 10^{-2} \text{ V} = 0.01 \text{ V} \]

### Step 4: Convert the induced emf to millivolts
To express the induced emf in millivolts (mV):
\[ \mathcal{E} = 0.01 \text{ V} = 10 \text{ mV} \]

### Final Answer
Thus, the induced emf in the circular loop at 0.25 seconds is:
\[ \mathcal{E} = 10 \text{ mV} \]