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Relation between induced charge and change in magnetic flux
According to the faraday’s law, the induced EMF is given as $$ {\mathcal {E}}=\frac{\Delta \phi}{\Delta t}$$ If $R$ is the total resistance of the closed-loop, the induced current will be $$I=\frac{\mathcal {E}}{R} \quad or \quad \frac{\Delta q}{\Delta t} =\frac{\Delta \phi}{\Delta t}\cdot\frac{1}{R}$$ Hence, the charge induced in the loop in time $\Delta t$ is
| $$\Delta q =\frac{\Delta \phi}{R}= \frac{\text{Net change in magnetic flux}}{\text{Resistance}}$$ |
Clearly, we can see the induced current depends on the net change in magnetic flux and does not depend on the time interval $\Delta t$ of the flux change. Thus, the induced charge does not depend on the rate of change of magnetic flux.
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