JEE Main & Advanced Physics Electro Magnetic Induction Periodic EMI

Suppose a rectangular coil having N turns placed initially in a magnetic field such that magnetic field is perpendicular to it?s plane as shown.

\[\omega -\] Angular speed

\[v-\] Frequency of rotation of coil

\[R-\]Resistance of coil

For uniform rotational motion with \[\omega ,\] the flux linked with coil at any time t

\[\varphi =NBA\cos \theta =NBA\cos \omega t\]

\[\phi ={{\phi }_{0}}\cos \,\omega t\] where \[{{\phi }_{0}}=NBA=\] maximum flux

(1) Induced emf in coil : Induced emf also changes in periodic manner that?s why this phenomenon called periodic EMI

\[e=-\frac{d\varphi }{dt}=NBA\omega \sin \omega \,t\]\[\Rightarrow \]\[e={{e}_{0}}\sin \,\omega t\] where \[{{e}_{0}}=emf\] amplitude or max. emf \[=NBA\omega ={{\varphi }_{0}}\omega \] 

(2) Induced current : At any time t, \[i=\frac{e}{R}=\frac{{{e}_{0}}}{R}\sin \omega \,t={{i}_{0}}\sin \omega \,t\] where \[{{i}_{0}}=\] current amplitude or max. current \[{{i}_{0}}=\frac{{{e}_{0}}}{R}=\frac{NBA\omega }{R}=\frac{{{\varphi }_{0}}\omega }{R}\]