In the circuit shown in the figure is assume that the operational amplifier is ideal and that $\mathrm{V}_{\mathrm{o}}=0 \mathrm{~V}$ initially. The switch is connected first to ' $A$ ' charging $C_{1}$ to the voltage $V$. It is then connected to the point ' $B$ '. This process is repeated $f$ times per second.
- Calculate the charge transferred per second from node $A$ to node $B$.
- Derive the average rate of change of the output voltage $\mathrm{V}_{\circ}$.
- If the capacitor and the switch are removed and a resistor is connected between points $A$ and $B$, find the value of the resistor to get the same average rate of change of the output voltage?
- If the repetition rate of the switching action is $10^{4}$ times per second, $C_{1}=100 \; p \mathrm{F}, C_{2}=10 \; p \mathrm{F}$ and $\mathrm{V}=10 \; \mathrm{mV}$, what is the average rate of change of the output voltage?