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 \( l \) moving with a constant velocity \( v \) towards the east in the presence of the Earth's magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a vertical rod of length \( l \) moving towards the east with velocity \( v \).
- The vertical component of the Earth's magnetic field is \( B \), and the angle of dip is \( \theta \).

2. **Identify the Components of the Magnetic Field**:
- The Earth's magnetic field can be resolved into two components:
- The vertical component \( B_v = B \) (given).
- The horizontal component \( B_h \) can be found using the angle of dip \( \theta \).
- From trigonometry, we know:
\[ \tan(\theta) = \frac{B_v}{B_h} \]
- Rearranging gives:
\[ B_h = \frac{B_v}{\tan(\theta)} = \frac{B}{\tan(\theta)} \]

3. **Calculate the Induced e.m.f.**:
- The formula for the induced e.m.f. (\( E \)) in a moving conductor in a magnetic field is given by:
\[ E = B_h \cdot l \cdot v \]
- Substituting \( B_h \):
\[ E = \left(\frac{B}{\tan(\theta)}\right) \cdot l \cdot v \]
- This simplifies to:
\[ E = \frac{B \cdot l \cdot v}{\tan(\theta)} \]

4. **Final Result**:
- The induced e.m.f. in the rod is:
\[ E = B \cdot l \cdot v \cdot \cot(\theta) \]

### Final Answer:
\[ E = B \cdot l \cdot v \cdot \cot(\theta) \]

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