Capacitors are used in DC circuits to provide "bursts of energy." Typical examples would be a capacitor to jump start a motor or a capacitor used to operate a camera's flash.

Charging

In the circuit shown below, when the switch is closed, charges immediately start flowing from the battery onto the plates of the capacitor. As the charge on the capacitor's plates increases, this transient current decreases; until finally, the current ceases to flow and the capacitor is fully charged. In the diagram shown below, the right plate of the capacitor would be positively charged and its left plate negatively charged since the plates are arbitrarily assigned as + and - according to their proximity to the nearest battery terminal.

Graphs of charge vs time and current vs time for charging capacitors are shown below. Mathematically, both of these graphs are exponential functions - current is an exponential decay, while charge is an exponential growth.

When the switch is initially closed, the capacitor does not have carry any charge and Kirchoff's loop rule would result in the equation:

- R- V_c = 0
= R
= / R

Where represents the initial, maximum current flowing off the battery onto the plates of the capacitor. However, this current is not steady. As time passes, more and more charges accumulate on the capacitor. This in turn increases the voltage across the capacitor, Q = CV. When the voltage of the capacitor equals the voltage of the battery, charges will cease to flow. The current decreases with time.

We will now derive the equation for the transient charge on the capacitor. In this derivation, i represents the transient current in the circuit as the capacitor charges, q represents the transient charge present on the capacitor, q(t) represents the charge at any given time, t, and C is the capacitance of the capacitor.

We will use as our initial conditions: t = 0, I = / R, and V_c = 0. Applying Kirchoff's Loop Rule we have

where

Substituting these values and solving for the expression dq/dt we get

Separating variables and integrating yields

In order to solve for q(t) we must extricate it from within its natural log expression .

Sincewe will now raise both sides of our equation to a power of e.

In the equation above, Ce represents the final total charge present on the plates based on Q = CV.

To calculate the equation for the transient current, we will use the fact that and differentiate the equation we just derived for q(t).

In these equations, the product of RC must have the units of time, since the function f(x) = e^x must be dimensionless. Let's investigate this relationship.

This product is called the RC time constant. It tell us when certain percentages of change have occurred. For example, when t = 1 RC,

Discharging

When the switch is closed in the circuit shown below, charges immediately start flowing off of the plates of the capacitor. As the charge on the capacitor's plates decreases, the current decreases; until finally, the current ceases to flow and the capacitor is fully discharged. 

Graphs of current vs time and charge vs time will both be decay functions since the current flowing through the resistor will fall off according to the flow of charge off of the capacitor's plates.

We will now derive the equation for the transient charge on the capacitor. In this derivation, i represents the transient current in the circuit as the capacitor charges, q represents the transient charge remaining on the capacitor, q(t) represents the charge at any given time, t, and C is the capacitance of the capacitor.

The initial conditions are that when t = 0, and I = V_c / R. Applying Kirchoff's Loop Rule we have

To calculate the equation for the transient current, we will use the fact that and differentiate the equation we just derived for q(t).