Why
Would You Want To Store Energy in Capacitors?
Storing energy is very important. You count on the energy stored
in your gas tank if you drove a car to school or work today. That's
an obvious case of energy storage. There are lots of other places
where energy is stored. Many of them are not as obvious as the gas
tank in a car. Here are a few.

You're reading this on
a computer, and the computer keeps track of the date and time. It
does that by keeping a small part of the computer running when you think
that the computer is turned off. There's a small battery that stores
the energy to keep the clock running when everything else is turned off.

If you own a stereo or
television that you have to plug into the wall plug, then you should realize
that the wall plug voltage becomes zero 120 times a second. When
that happens, the system keeps running because there are capacitors inside
the system that store energy to carry you through those periods when the
line voltage isn't large enough to keep things going!
Capacitors can't really be used to store a lot of energy, but there are
many situations in which a capacitor's ability to store energy becomes
important. In this lesson we will discuss how much energy a capacitor
can store. Your goal for this lesson is this.
Capacitors are often used to store energy.

When relatively small
amounts of energy are needed.

Where batteries are not
desired because they might deteriorate.

For larger power/short
duration applications  as in power supply filters, or to keep power up
long enough for a computer to shut down gracefully when the line power
fails.
How
Much Energy Can Be Stored In A Capacitor?
To calculate how much energy is stored in a capacitor, we start by looking
at the basic relationship between voltage and current in a capacitor.
First, here is the circuit symbol for a capacitor with the current into
the capacitor and the voltage across it defined.
Then, the relationship between the current
and the voltage in the capacitor is given by:
i(t)
= C dv(t)/dt
Once we have this relationship, we can calculate the power  the rate of
flow of energy into the capacitor  by multiplying the current flowing
through the capacitor by the voltage across the capacitor. And, once
we have the power  the rate of flow of energy into the capacitor  we
can eventually calculate the energy stored in the capacitor. We know:
P(t) = i(t)v(t)

Given the expression for
the power:

And given the expression
for the current:

We can use the expression
for current in the power expression:

And, we can recognize
that power is simply rate of energy input.

Now, the derivative of
energy can be integrated to find the total energy input. Setting
up the integral we have:

Then, integrating we have:

After integrating we have:

or simply:
Some
Things To Note About The Energy Storage Formula
Note the following about the energy stored in the capacitor.

The energy stored in a
capacitor is proportional to the capacitance.

The energy stored in a
capacitor is proportional to the square of the voltage across the capacitor.

The expression for the
energy stored in a capacitor resembles other energy storage formulae.

For kinetic energy, with
a mass, M, and a velocity, v, the stored energy is E
= (Mv^{2}/2)

For potential energy,
with a spring constant, K, and an elongation, x, the stored energy is E
= (Kx^{2}/2)
Note also the following
observations.
And some final points
to note.

Power in to the capacitor
can be negative.

Voltage can be positive
while current is negative.

Imagine a capacitor that
is charged. You could charge a capacitor by putting a battery across
the capacitor, for example. Then, if you placed a resistor across
the capacitor, charge would leave the capacitor  current would flow out
of the capacitor  and the energy in the capacitor would leave the capacitor
only to become heat energy in the resistor.

When energy leaves the
capacitor, power is negative.

When you use capacitors
in a circuit and you analyze the circuit you need to be careful about sign
conventions as defined in the illustration we used above  which is repeated
here.
