A flip coil of enclosed area A consists of $N$ turns of circular coils which lie in a uniform magnetic field. Plane of the coils is perpendicular to the magnetic field as shown in Fig. The coil is connected to a current integrator which measures the total charge passing through it. The coil is turned through $180°$ about its diameter. Find the charge passing through the coil.

When the flip coil is rotated, the magnetic flux through it changes, resulting in an induced emf $E$ and as induced current $I=E/R$, where $R$ is the total resistance of the circuit.

As $I=\frac{dQ}{dt}$

    $Q=\int dQ=\int Idt$

From Faraday's law, induced emf 

$\left|E\right|=\frac{d{\varphi }_B}{dt}$

Thus $Q=\int Idt=\int \frac{E}{R}dt=\frac{1}{R}\int \frac{d{\varphi }_B}{dt}dt$

                        $=\frac{1}{R}\int d{\varphi }_B=\frac{\Delta {\varphi }_B}{R}$

This is a generalized result and can be used to determine charge through a circuit in which an emf and a current is induced.

Initial flux through the coil, ${\varphi }_{Bi}=+NBA$.

Final flux through the coil, ${\varphi }_{Bf}=-NBA$.

When the coil is turned through $180°$ its flux reverses, angle between magnetic field and area vector is reversed. 

$\Delta {\varphi }_B={\varphi }_{Bf}-{\varphi }_{Bi}=NBA-(-NBA)=2NBA$

$Q=\frac{2NBA}{R}$  Ans.