Rotating Rod in Magnetic Field
Example 6.6: EMF in a Rotating Rod
A metallic rod of 1 m length rotates at 50 rev/s, with one end hinged at the center and the other end at the circumference of a circular metallic ring of radius 1 m. A uniform magnetic field of 1 T parallel to the axis is present everywhere.
Current frequency: 50 rev/s
Current field: 1.0 T
Length of rod (R): 1.0 m
Angular velocity (ω): 314.16 rad/s
Calculated EMF (ε): 157.08 V
Method I: Lorentz Force Calculation
As the rod rotates, free electrons move towards the outer end due to Lorentz force, creating an emf:
dε = Bv dr
ε = ∫ Bωr dr = ½BωR²
ε = ½ × 1.0 × (2π × 50) × (1)² = 157 V
Method II: Rate of Change of Area
EMF equals B × (rate of change of area of loop):
Area = ½R²θ
ε = B × d/dt[½R²θ] = ½BR²ω
Same result as Method I: ε = 157 V
