# Time Constant of a Capacitor in DC

I've just started studying Physics and I'm not sure how you're supposed to calculate the time constant of a capacitor when a direct current is applied. DC means that frequency equals = so the resistance is infinite, but you need a value for the resistance to use the formula tau = RC. I just did the experiment so I know it doesn't take an infinite amount of time for the capacitor to discharge. I've tried Google a lot but they only mention calculating the time constant for alternating current.

Thanks in advance :)

• Time constant doesn't depend on frequency and it doesn't depend on the capacitor's "resistance" (actually impedance). – user253751 Sep 22 '20 at 11:52
• Well impedance is equal to 1/2*pifC. f is equal to 0 so it's still infinite. And the formula I was given mentions resistance as I stated in the question, tau = RC – Jordan Sep 22 '20 at 11:55
• Yes. You will notice that the impedance isn't the resistance. You might also notice that capacitors don't have time constants, RC circuits do. You have the C, where's the R? – user253751 Sep 22 '20 at 11:58
• The time constant of a capacitor is the time it takes to charge 63% of its maximum charge. Given by the formula t = RC. That was provided by my lecturer. I don't know where the R comes from, that's why I'm asking this question. – Jordan Sep 22 '20 at 12:02
• Well obviously you can't calculate how long it takes to charge unless you know what's charging it. What's charging it? If you charge a capacitor with an ideal current source, the maximum charge is infinite (real capacitors will explode). – user253751 Sep 22 '20 at 12:49

## 1 Answer

I'm not sure how you're supposed to calculate the time constant of a capacitor when a direct current is applied.

The time constant for the capacitor is simply RC and it applies to both AC and DC circuits, but only under transient conditions such as during a period of time just after a switch connects or disconnects a capacitor to the circuit. After a long time (steady state conditions), the RC time constant is not involved in either an AC or DC circuit.

DC means that frequency equals = 0, the resistance is infinite, but you need a value for the resistance to use the formula tau = RC.

The $$R$$ in the time constant is not the resistance of the capacitor, but the resistance of a resistor connected to the capacitor. While it is true that the impedance (the term impedance applies here, not resistance) of an ideal capacitor is infinite at zero frequency, during transient conditions currents an voltages are changing so the impedance of the capacitor is not infinite under transient conditions. The applicable general equations for voltage and current in an ideal capacitor when voltages and currents are changing are

$$i(t)=C\frac{dv(t)}{dt}$$

$$v(t)=\frac{1}{C}\int i(t)dt$$

Below is a series RC DC circuit. At time $$t=0$$ the switch is closed. The currents and voltages in the circuit as a function of time assuming the initial voltage across the capacitor before the switch is closed is zero are

$$v_{C}(t)=V(1-e\large^{-\frac{t}{RC}})$$

$$i(t)=\frac{V}{R} {e\large^{-\frac{t}{RC}}}$$

$$v_{r}(t)=i(t)R=V{e\large^{-\frac{t}{RC}}}$$

Note that the time constant is RC.

When $$t=0$$ the voltage across the capacitor is zero, the current in the circuit is the maximum of $$V/R$$ .

When $$t=∞$$. The current is zero and the voltage across the capacitor is the battery voltage $$V$$.

Hope this helps. • Thank you so much. Very helpful explanation. – Jordan Sep 22 '20 at 23:08
• You are very welcome and I’m glad it helped – Bob D Sep 22 '20 at 23:28