A vertical rod of length l is moved with constant velocity v towards east. The vertical component of earth magnetic field is B and angle of dip is `theta`. The induced e.m.f. in the rod is

Options

A. `Blv sin theta`
B. `Blv tan theta`
C. `Blv cot theta`
D. `Blv cos theta`

Answer

To find the induced electromotive force (e.m.f.) in a vertical rod of length \( l \) that is moved with a constant velocity \( v \) towards the east, we will use the principles of electromagnetic induction. The vertical component of the Earth's magnetic field is \( B \), and the angle of dip is \( \theta \).

### Step-by-Step Solution:

1. **Identify the Components of the Magnetic Field**:
The Earth's magnetic field can be resolved into two components due to the angle of dip \( \theta \):
- The vertical component \( B_v = B \cos(\theta) \)
- The horizontal component \( B_h = B \sin(\theta) \)

2. **Determine the Horizontal Component**:
Since we are interested in the horizontal component of the magnetic field for the induced e.m.f. when the rod moves eastward:
\[ B_h = B \sin(\theta) \]

3. **Use the Formula for Induced e.m.f.**:
The induced e.m.f. (\( \mathcal{E} \)) in a rod moving through a magnetic field is given by the formula:
\[ \mathcal{E} = B_h \cdot l \cdot v \]
where:
- \( B_h \) is the horizontal component of the magnetic field,
- \( l \) is the length of the rod,
- \( v \) is the velocity of the rod.

4. **Substituting the Horizontal Component**:
Substitute \( B_h = B \sin(\theta) \) into the e.m.f. formula:
\[ \mathcal{E} = (B \sin(\theta)) \cdot l \cdot v \]

5. **Final Expression for Induced e.m.f.**:
Thus, the induced e.m.f. in the rod can be expressed as:
\[ \mathcal{E} = B \cdot l \cdot v \cdot \sin(\theta) \]

### Conclusion:
The induced e.m.f. in the vertical rod is given by:
\[ \mathcal{E} = B \cdot l \cdot v \cdot \sin(\theta) \]