A wheel of 20 metallic spokes each 40 cm long is rotated with a speed of 180 rev/min in a plane normal to the horizontal component of earth’s magnetic field $H_e$ at a place. If $H_e=0.4G$  (Gauss) at that place, the induced emf between the axle and the rim of the wheel is 

Induced EMF Solution

Explanation and Solution for the Question

We are tasked with calculating the induced EMF between the axle and the rim of a wheel with 20 metallic spokes. Here's how we can solve the problem step by step:

Step 1: Understand the Given Data

• Number of spokes: N = 20 (though it is not directly relevant for EMF calculation since we deal with one spoke).
• Length of each spoke: L = 40 cm = 0.4 m.
• Rotational speed: f = 180 rev/min = 3 rev/s.
• Horizontal component of Earth’s magnetic field: H_e = 0.4 G = 0.4 × 10^-4 T.

Step 2: Formula for Induced EMF

The EMF induced in a rotating wheel with metallic spokes is given by:

EMF = (1/2) B ω L²

Where:

• B: Magnetic field strength (H_e here).
• ω: Angular velocity in radians per second.
• L: Length of a metallic spoke.

Step 3: Convert Rotational Speed to Angular Velocity

The angular velocity ω is related to the frequency f by:

ω = 2πf

Substitute f = 3 rev/s:

ω = 2π × 3 = 6π rad/s

Step 4: Substitute Values into the Formula

Using B = 0.4 × 10^-4 T, ω = 6π rad/s, and L = 0.4 m:

EMF = (1/2) × B × ω × L²

Substitute the values:

EMF = (1/2) × (0.4 × 10^-4) × (6π) × (0.4)²

1. Calculate L²: (0.4)² = 0.16 m².
2. Multiply: 0.16 × 6π = 0.96π.
3. Multiply: 0.96π × 0.4 × 10^-4 = 0.384π × 10^-4.
4. Multiply by (1/2): (1/2) × 0.384π × 10^-4 = 0.192π × 10^-4.

Step 5: Final Answer

The induced EMF between the axle and the rim of the wheel with 20 metallic spokes is:

EMF = 192π × 10^-7 V

Correct Option: a) 192π × 10^-7 V