A player with 3 meter long iron rod runs toward east with a speed of `30km//hr` . Horizontal component of eath's magnetic field is `4xx10^(-5)Wb//m^(2)` . If he runs with the rod in horizontal and vertical position, then the potential difference induced between the two ends of the rod in the two cases will be

Options

A. Zero in vertical position and `1 xx 10^(-3)` V in horizontal position
B. `1 xx 10^(-3)` V in vertical position and zerp is horizontal position
C. Zero in both cases
D. ` 1 xx 10^(-3)` V in both cases

Answer

To solve the problem of the induced potential difference in the iron rod when it is running horizontally and vertically, we can follow these steps:

### Step 1: Convert the speed from km/hr to m/s
The speed of the player is given as 30 km/hr. To convert this to meters per second (m/s), we use the conversion factor:
\[ 1 \text{ km/hr} = \frac{1}{3.6} \text{ m/s} \]
Thus,
\[ 30 \text{ km/hr} = \frac{30}{3.6} \text{ m/s} \approx 8.33 \text{ m/s} \]

### Step 2: Identify the magnetic field and rod length
The horizontal component of the Earth's magnetic field is given as:
\[ B = 4 \times 10^{-5} \text{ Wb/m}^2 \]
The length of the rod is:
\[ L = 3 \text{ m} \]

### Step 3: Analyze the two positions of the rod
1. **Horizontal Position**: When the rod is held horizontally, the angle \( \theta \) between the velocity vector and the magnetic field is 90 degrees (since the rod is moving east and the magnetic field is directed north). The induced EMF (\( \mathcal{E} \)) can be calculated using the formula:
\[ \mathcal{E} = B \cdot L \cdot v \cdot \sin(\theta) \]
Here, \( \sin(90^\circ) = 1 \):
\[ \mathcal{E}_{\text{horizontal}} = B \cdot L \cdot v = (4 \times 10^{-5}) \cdot (3) \cdot (8.33) \]
\[ \mathcal{E}_{\text{horizontal}} = 1 \times 10^{-3} \text{ V} = 1 \text{ mV} \]

2. **Vertical Position**: When the rod is held vertically, the angle \( \theta \) between the velocity vector and the magnetic field is 0 degrees. Thus:
\[ \mathcal{E}_{\text{vertical}} = B \cdot L \cdot v \cdot \sin(0^\circ) = 0 \]
Therefore, no EMF is induced when the rod is vertical.

### Final Results
- Induced potential difference when the rod is horizontal: \( 1 \text{ mV} \)
- Induced potential difference when the rod is vertical: \( 0 \text{ V} \)

### Conclusion
The potential difference induced between the two ends of the rod is:
- **Horizontal position**: \( 1 \times 10^{-3} \text{ V} \)
- **Vertical position**: \( 0 \text{ V} \)

Thus, the correct option is **Option 2**: \( 1 \times 10^{-3} \text{ V} \) in vertical position and \( 0 \) in horizontal.