A metal wheel with 20 metallic spokes each 1m long, is rotated with a speed of 180 rpm, in a plane perpendicular to the earth's field. at that place. If the magnitude of the field is 0.40 gauss, then the emf between the axle and rim of the wheel is 

CONCEPT:

• Motional emf: The emf induced due to motion relative to a magnetic field is called the motional emf. 

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We have:

• Number of spokes (N) = 20

• Length of each spoke (r) = 1 m

• Speed of rotation (n) = 180 rpm

• Magnetic field (B) = 0.4 G = 0.4 × 10^-4 T (since 1 G = 10^-4 T)

The angular velocity (ω) in radians per second is:
ω = 2πn / 60

⇒ ω = 2π × 180 / 60 = 6π rad/s

The induced emf (ε) is given by Faraday's law:

ε = -dΦ/dt

Magnetic flux (Φ) is given by:

Φ = B · A

Area swept by one spoke in one second:
A = (1/2) · r² · θ, where θ = ω · t = 6π · 1 = 6π

So, A = (1/2) · r² · 6π = 3π · r²

Substituting r = 1 m:

A = 3π · (1)² = 2π m²

Φ = B · A = 0.4 × 10^-4 · 3π

ε = (1/2) · B · ω · r²

= (1/2) · (0.4 × 10^-4) · (6π) · (1)²

= 0.2 × 10^-4 · 6π = 1.2π × 10^-4 V

Using π ≈ 3.14:

ε ≈ 1.2 · 3.14 × 10^-4 ≈ 3.77 × 10^-4 V

ε ≈ 3.7 × 10^-4 V