How can i show that the magnitude of the charge on a capacitor is given by...?

Question

Q=Q0(1-e to the square root -t/pie) and Q=Qo e to the square root -t/pie for charging and discharging? In an RC circuit

Answer 1

Your equations are not quite right, at least the one for charging, because the final charge as time t approaches infinity depends on the capacitance and the source voltage, neither of which are in your equation. Nor do the square root or π belong in the equations, but the time constant τ (tau, which equals RC) is missing.

When charging to a source voltage Vs, the voltage V across a capacitor in a simple RC circuit is simply:
V = Vo + [Vs-Vo][1 - exp (-t/τ)]
where Vo is the voltage at time zero, and exp ( ) is the notation for e to the power shown in the parentheses.

The discharge equation is even simpler because the capacitor discharges to zero:
V = Vo [exp (-t/τ)]

Since charge equals voltage times capacitance (Q=CV), the above equations can be modified to give:
Q = Qo + [CVs-Qo][1 - exp (-t/τ)], and
Q = Qo [exp (-t/τ)]