Question

Show that the total induced charge simply depends upon the change in magnetic Flux and is independent of the time rate of change of flux

Answer 1

we know induced Emf  E=-df/dt                      (lets consider f denotes magnetic flux)
   current I=E/R           where R is the resistance
now I=dq/dt where q is the charge
therefore equating, dq/dt=-df/dt R
cancelling dt on both sides, we get dq=-df/R
therefore we can say that induced charge does not depend on rate of change of flux

Answer 2

We know that the induced emf E = -$\frac{d phi}{dt}$   ...(i)

where phi is the magnetic flux

Current I = $\frac{E}{R}$       ....(ii)

where R is resistance

I = $\frac{dq}{dt}$      ....(iii)

Putting value of (i) in (ii)

we get I = -$\frac{\frac{d phi}{dt}}{R}$       ....(iv)

Equating (iii) with (iv)

$\frac{dq}{dt}$ = -$\frac{\frac{d phi}{dt}}{R}$

dq = -$\frac{d phi}{R}$

We can say from the above equation that the induced charge does not depend on rate of change of flux.

Hence proved mathematically