A rod AB slides on a V shaped wire with speed v as shown, such that at any time OA = OB = I. Magnetic field in the region is perpendicular downwards and has strength B, induced emf in the rod is

Options

A. Zero
B. Bvl
C. `(sqrt(3)Bvl)/(2)`
D. `(Bvl)/(2)`

Answer

To find the induced electromotive force (emf) in the rod AB that slides on a V-shaped wire, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a rod AB that is sliding on a V-shaped wire. The rod moves with a speed \( v \).
- The lengths OA and OB are equal and denoted as \( L \).
- The magnetic field \( B \) is directed downwards, perpendicular to the plane of the wire.

2. **Identifying the Geometry**:
- The angle of the V-shaped wire is \( 60^\circ \).
- The rod AB can be considered as part of an isosceles triangle where OA = OB = L.

3. **Using the Formula for Induced EMF**:
- The induced emf (\( E \)) in a conductor moving through a magnetic field can be calculated using the formula:
\[ E = \int (v \cdot B) \, dl \]
- Here, \( v \) is the velocity of the rod, \( B \) is the magnetic field strength, and \( dl \) is the differential length element along the rod.

4. **Simplifying the Expression**:
- Since the rod is moving with a constant speed \( v \) and the magnetic field \( B \) is uniform, we can simplify the integral:
\[ E = B \cdot v \cdot L \]
- Here, \( L \) is the effective length of the conductor in the magnetic field.

5. **Conclusion**:
- The induced emf in the rod AB is given by:
\[ E = B v L \]

### Final Answer:
The induced emf in the rod is \( E = B v L \).

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