A Conducting ring of radius 1 meter is placed in an uniform magnetic field `B` of `0.01` tesla oscillating with frequency `100 Hz` with its plane at right angles to `B`. What will be the induced electric field?

Options

A. `pi` volts/m
B. 2 volts/m
C. 10 volts/m
D. 62 volts/m

Answer

To solve the problem of finding the induced electric field in a conducting ring placed in an oscillating magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of the conducting ring, \( r = 1 \) meter
- Magnetic field strength, \( B = 0.01 \) tesla
- Frequency of oscillation, \( f = 100 \) Hz

2. **Calculate the Area of the Ring**:
The area \( A \) of the ring can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 = \pi (1^2) = \pi \, \text{m}^2 \]

3. **Determine the Time Period**:
The time period \( T \) is the reciprocal of the frequency:
\[ T = \frac{1}{f} = \frac{1}{100} = 0.01 \, \text{s} \]

4. **Calculate the Change in Magnetic Flux**:
The magnetic flux \( \Phi \) through the ring is given by:
\[ \Phi = B \cdot A \]
Since the magnetic field is oscillating, we consider the maximum magnetic flux:
\[ \Phi_{\text{max}} = B \cdot A = 0.01 \cdot \pi \]

5. **Calculate the Induced EMF**:
The induced EMF \( \mathcal{E} \) can be calculated using Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Since the magnetic field oscillates sinusoidally, the average rate of change of flux over one quarter of the cycle (from 0 to maximum) is:
\[ \mathcal{E} = \frac{4 \Phi_{\text{max}}}{T} = \frac{4 \cdot (0.01 \cdot \pi)}{0.01} = 4\pi \, \text{volts} \]

6. **Relate Induced EMF to Electric Field**:
The induced EMF is also related to the electric field \( E \) around the ring:
\[ \mathcal{E} = E \cdot (2\pi r) \]
Substituting \( r = 1 \):
\[ 4\pi = E \cdot (2\pi) \]

7. **Solve for the Electric Field**:
Rearranging the equation gives:
\[ E = \frac{4\pi}{2\pi} = 2 \, \text{volts/m} \]

### Final Answer:
The induced electric field in the conducting ring is \( \mathbf{E = 2 \, \text{volts/m}} \).

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