Rotating rods and wheels

A rod of length R turning at angular speed ω about one end, in a uniform field B along the axis, has different parts moving at different speeds v = ωr.

Adding the emf B v dr from each small piece gives ε = ½BωR² between the pivot and the free end.

The same result follows from Faraday's law: the rod sweeps a sector of area ½R²θ, and B times the rate of change of that area is ½BR²ω.

Example 6.6: a 1 m rod turning at 50 rev/s in a 1 T field gives ε = ½ × 1.0 × 2π × 50 × 1² = 157 V between the centre and the rim.

Example 6.7: a wheel with 10 spokes of 0.5 m, turning at 120 rev/min in the earth's horizontal field of 0.4 G, gives ε = ½ × 4π × 0.4 × 10⁻⁴ × 0.5² = 6.28 × 10⁻⁵ V between axle and rim.

The number of spokes does not matter: the spokes are emfs in parallel, each giving the same value.