A 1.0 m metallic rod is rotated with an angular velocity of 400 rad/s about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant magnetic field of 0.5 T parallel to the axis every where Calculate the emf developed between the centre and the ring.

Options

A. 99V
B. 100 V
C. 102 V
D. 105 V

Answer

To solve the problem of calculating the emf developed between the center and the ring of a rotating metallic rod in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the parameters given in the problem:**
- Length of the rod (L) = 1.0 m
- Angular velocity (ω) = 400 rad/s
- Magnetic field (B) = 0.5 T

2. **Understand the setup:**
- The rod is rotating about one end, and the other end is in contact with a circular metallic ring.
- The magnetic field is parallel to the axis of rotation.

3. **Use the formula for induced emf (E):**
- The induced emf in a rotating rod can be calculated using the formula:
\[ E = \frac{1}{2} B \omega L^2 \]
- Here, \(L\) is the length of the rod.

4. **Substitute the values into the formula:**
- Plugging in the values:
\[ E = \frac{1}{2} \times 0.5 \, \text{T} \times 400 \, \text{rad/s} \times (1.0 \, \text{m})^2 \]

5. **Calculate the emf:**
- First, calculate the product:
\[ E = \frac{1}{2} \times 0.5 \times 400 \times 1 = \frac{1}{2} \times 200 = 100 \, \text{V} \]

6. **Conclusion:**
- The emf developed between the center and the ring is **100 volts**.