Derive the formula for induced charge and prove that it is independent of the rate of change of magnetic flux.

(N/A) According to Faraday's law of electromagnetic induction,the magnitude of the induced electromotive force (emf) is given by:
$|\varepsilon| = \frac{\Delta \Phi_{B}}{\Delta t} \quad \dots (1)$
We also know that the induced current $I$ is related to the induced emf by Ohm's law,$|\varepsilon| = I r$,where $r$ is the resistance of the coil. Since $I = \frac{\Delta Q}{\Delta t}$,we can write:
$|\varepsilon| = \frac{\Delta Q}{\Delta t} r \quad \dots (2)$
Equating $(1)$ and $(2)$:
$\frac{\Delta \Phi_{B}}{\Delta t} = \frac{\Delta Q}{\Delta t} r$
Canceling $\Delta t$ from both sides,we get:
$\Delta \Phi_{B} = \Delta Q \cdot r$
Therefore,the induced charge is:
$\Delta Q = \frac{\Delta \Phi_{B}}{r}$
Since the expression for $\Delta Q$ depends only on the total change in magnetic flux $\Delta \Phi_{B}$ and the resistance $r$,it is independent of the time interval $\Delta t$ and consequently independent of the rate of change of flux $\frac{\Delta \Phi_{B}}{\Delta t}$.