Question

A 4.0 µF capacitor is connected in series with a 2.0 kΩ resistor across a 20-V DC source and an open switch. If the switch is closed at t = 0 s, what is the charge on the capacitor at t = 9 ms?

Answer 1

Do you know what the Voltage-Current relationship for a capacitor is? Have you solved the differential equation for a capacitor and a resistor in series?
This equation gives the current for a capacitor in terms of its voltage:
\[I_c = C \frac{ dV_c }{ dt }\]
Because the resistor and capacitor are in series, the same current will flow through both. For t> 0 there is a circuit consisting of the 20V source, the resistor, and the capacitor, and all the voltage drops around the circuit must sum to zero. If you write that in equation form you have the differential equation for t>0.

The initial condition is that the capacitor voltage is zero when t= 0. I think you should assume this because the problem does not mention any initial charge on the capacitor.

Answer 2

Rather I should have asked "Have you solved the differential equation for a capacitor and a resistor in series, with a constant voltage source across it?"