# A stiff semi-circular wire of radius R is rotated in a uniform magnetic field B about an axis passing through its ends.

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A stiff semi-circular wire of radius R is rotated in a uniform magnetic field B about an axis passing through its ends. If the frequency of rotation of the wire is f, calculate the amplitude of the alternating emf induced in the wire.

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In one rotation, the wire traces out a circle of radius R, i.e., an area A = πR2.
Therefore, the rate at which the wire traces out the area is
$$\frac{d A}{d t}$$ = frequency or rotation × A = fA
If the angle between the uniform magnetic field $$\vec{B}$$ and the rotation axis is θ, the magnitude of the induced emf is
|e|= B$$\frac{d A}{d t}$$ cosθ = BfA cosθ = Bf(πR2)cosθ
so that the required amplitude is equal to Bf(πR2)