Deternine the magnitude of the emf generated between the ends of the axle of a railway carriage, 1m in length, when it is moving with a velocity of 36 km/hr along a horizontal track, given horizontal component to Earth's magnetic field `B_H=4 xx 10^(-5)` Tesla and angle of dip `delta= 60^@`.

Options

A. `7.9 xx 10^(-4)` volt
B. `6.9 xx 10^(-4)` Volt
C. `5.9 xx 10^(-4)` Volt
D. `3.9 xx 10^(-4)` Volt

Answer

To determine the magnitude of the electromotive force (emf) generated between the ends of the axle of a railway carriage, we will follow these steps:

### Step 1: Convert the velocity from km/hr to m/s
The given velocity of the railway carriage is 36 km/hr. To convert this to meters per second (m/s), we use the conversion factor \( \frac{5}{18} \).

\[ v = 36 \, \text{km/hr} \times \frac{5}{18} = 10 \, \text{m/s} \]

### Step 2: Identify the components of the Earth's magnetic field
We are given the horizontal component of the Earth's magnetic field \( B_H = 4 \times 10^{-5} \, \text{T} \) and the angle of dip \( \delta = 60^\circ \). The vertical component \( B_V \) can be calculated using the relationship:

\[ B_V = B_H \tan(\delta) \]

Calculating \( B_V \):

\[ B_V = 4 \times 10^{-5} \, \text{T} \times \tan(60^\circ) = 4 \times 10^{-5} \, \text{T} \times \sqrt{3} \approx 4 \times 10^{-5} \times 1.732 = 6.928 \times 10^{-5} \, \text{T} \]

### Step 3: Calculate the induced emf using the formula for motional emf
The formula for the motional emf \( \mathcal{E} \) induced in a conductor moving through a magnetic field is given by:

\[ \mathcal{E} = B \cdot L \cdot v \]

Where:
- \( B \) is the vertical component of the magnetic field \( B_V \)
- \( L \) is the length of the axle (1 m)
- \( v \) is the velocity (10 m/s)

Substituting the values:

\[ \mathcal{E} = B_V \cdot L \cdot v = (6.928 \times 10^{-5} \, \text{T}) \cdot (1 \, \text{m}) \cdot (10 \, \text{m/s}) \]

Calculating \( \mathcal{E} \):

\[ \mathcal{E} = 6.928 \times 10^{-5} \cdot 10 = 6.928 \times 10^{-4} \, \text{V} \]

### Step 4: Final Result
Thus, the magnitude of the emf generated between the ends of the axle is:

\[ \mathcal{E} \approx 6.93 \times 10^{-4} \, \text{V} \text{ or } 693 \, \mu\text{V} \]