A rectangular coil of area 'A', having number of turns N, is rotated at 'f' revolutions per second in a uniform magnetic field B, the field being perpendicular to the coil. Prove that the maximum emf induced in the coil is `2 pi fNBA.`

Explanation

Consider a rectangular coil PQRS of N turns, each of area A, held in a uniform magnetic field `vecB`. Let the coil be rotated at a steady angular velocity w about its own axis.
Let at any instant t, normal to the area (i.e., the area vector `vecA`) subtends an angle `theta = omega t` from direction of magnetic field `vecB`. Then, at that moment, the magnetic flux linked with the coil is
` phi_(B) = NvecB.vecA = NB Acos theta = NB Acos omegat`
`therefore` Induced emf `varepsilon = - (dphi_(B))dt = - d/dt (N B A cos omegat)`
` =-N B A d/dt(cos omegat) = N B A omegat`
where `varepsilon_(0) = N B A omega` = maximum value of induced emf.
If the coil be rotating at .f. revolutions per second, then `omega = 2pif` and the maximum induced emf
`varepsilon_(0) = NBAomega = 2pifNBA`