For a place the Earth's magnetic field is `4 xx 10^(-5) T` and angle of dip is `60^(@)`. A wheel of radius 60 cm having 10 spokes is rotating with a speed of `150 "rev" // "min"` in plane perpendicular to Earth's horizontal magnetic field. Find the induced emf between axle and rim of the wheel.

Answer

To find the induced EMF between the axle and rim of the rotating wheel, we can follow these steps:

### Step 1: Determine the Horizontal Component of Earth's Magnetic Field
Given:
- Total Earth's magnetic field, \( B = 4 \times 10^{-5} \, \text{T} \)
- Angle of dip, \( \delta = 60^\circ \)

The horizontal component of the magnetic field \( B_H \) can be calculated using the formula:
\[ B_H = B \cos(\delta) \]

Substituting the values:
\[ B_H = 4 \times 10^{-5} \cos(60^\circ) = 4 \times 10^{-5} \times \frac{1}{2} = 2 \times 10^{-5} \, \text{T} \]

### Step 2: Convert the Rotational Speed to Radians per Second
Given:
- Rotational speed = 150 revolutions per minute (rev/min)

To convert this to radians per second:
\[ \omega = 150 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = 5\pi \, \text{rad/s} \]

### Step 3: Calculate the Length of the Spoke
Given:
- Radius of the wheel \( r = 60 \, \text{cm} = 0.6 \, \text{m} \)

The length of one spoke \( L \) is equal to the radius:
\[ L = 0.6 \, \text{m} \]

### Step 4: Use the Induced EMF Formula
The formula for the induced EMF \( E \) in a rotating wheel is given by:
\[ E = \frac{1}{2} B_H L^2 \omega \]

Substituting the values:
\[ E = \frac{1}{2} \times (2 \times 10^{-5}) \times (0.6)^2 \times (5\pi) \]

Calculating:
\[ E = \frac{1}{2} \times 2 \times 10^{-5} \times 0.36 \times 5\pi \]
\[ E = 10^{-5} \times 0.18 \times 5\pi \]
\[ E \approx 10^{-5} \times 0.18 \times 15.70796 \approx 2.82 \times 10^{-4} \, \text{V} \]

### Step 5: Final Calculation
Calculating the final value:
\[ E \approx 5.65 \times 10^{-5} \, \text{V} \]

### Conclusion
The induced EMF between the axle and rim of the wheel is approximately:
\[ E \approx 5.65 \times 10^{-5} \, \text{V} \]