Show mathematically that the rotation of a coll in a magnetic field over one rotation Induces an alternating emf of one cycle.

Explanation

Let us consider a rectangular coil of N turns kept in a uniform magnetic field `vesB`.
The coil rotates in anti-clockwise direction with an angular velocity `omega` about an axis, perpendicularto the field.

At time=0, the plane of the coil is perpendicular to the field and the flux linked with the coil has its maximum value `phi_(m)`=BA (where A is the area of the coil ).
In a time t seconds, the coil is rotated through an angle ` theta(=omegat)` in anti-clockwish direction. In the position, the flux linked is `phi_(m) cos omegat`, a component of `phi_(m)` normal to the plane of the coil.
The component paralled to the plane `(phi_(m)sin omegat)` has no role in electromagnetic inducion. Therefore, the flux linkage at this deflected position is
`Nphi_(B)=Nphi_(m)cosomegat`