Capacitors are used in DC circuits to provide "bursts of energy." Typical examples would be a capacitor used to jump start a motor or a capacitor used to charge a camera's flash or a capacitor used to provide a large voltage across the chest in an emergency defibrillator when a person suffers from cardiac arrest.

 

 

 

When the switch is closed, charges immediately start flowing 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 above, 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 current vs time and charge vs time are shown below. Mathematically, both of these graphs are exponential functions - current is an example of exponential decay, while charge is an example of exponential growth.

 

Charging Capacitor Graphs
current vs time	charge vs time

 

 

At t = 0 seconds, when the switch is initially closed, the capacitor does not have carry any charge and Kirchoff's loop rule would result in the equation:

 

 

where I_max 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 equations for the transient charge on the capacitor and the transient current in the circuit. In this derivation,

 

 

Applying Kirchoff's Loop Rule we have

 

 

Separating variables and integrating both sides yields

 

 

In order to solve for q(t) we must first extricate it from within its natural log expression: . To do this, we will raise both sides of our equation to a power of e since .

 

 

In the equation above the final total charge present on the plates is. This allows us to rewrite our final equation for the build up of charge on the capacitor's plates as

     

 

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

 

t	RC
sec	ohms (farads) (volts/amps) (coulombs/volts) coulombs/amps coulombs/(coulombs/sec) sec

 

This product is called the RC time constant and it allows us the ability to determine when certain percentages of change have occurred. To calculate the equation for the transient current, we will use the fact that  and differentiate the equation we just derived for q(t).

 

   

 

Let's use this information to work an example of a charging capacitor.

 

 

 

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.

 

 

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

 

Discharging Capacitor Graphs
current vs time	charge vs time

 

We will now derive the equation for the transient charge on the capacitor. In this derivation,

 

 

Once again, by applying Kirchoff's Loop Rule, separating variables and integrating, we have

 

 

 

To calculate the equation for the transient current we must differentiate the equation we just derived for charge and use the fact that the charge on the capacitor is decreasing, .

 

Let's practice once again, but with discharging capacitors.