A capacitor discharge circuit of time constant 45ms includes a capacitor and resistor. The capacitor has a capacitance of 18µF What is the resistance of the resistor?

This question is in one of the CIE Physics Pre-U sample papers, and if you know the definition of the time constant for a capacitor circuit (=RC), its very easy, however even if you didnt it canbe derived fairly easily.

Consider a circuit containing just a capacitor C and resistor R, with the capacitor initially storing some charge Q0 with a voltage V0 across it. Call the curent flowing i, the voltage across the capacitor as time progresses Vc and the changing charge Q. By Kirchoff's law,

Vc + iR=0

Then we know that dQ/dt = i and C=Q/Vc, so we can say:

Q/C + RdQ/dt=0 and so RCdQ/dt + Q=0 

This has solution Q=Q0exp(-t/RC) and comparing this with the standard form of a time decaying property, X=X0exp(-t/k), where k is the time constant, gives k=RC, and then the problem is trivial.

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