Today’s agenda:

Energy Storage in Capacitors.You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered.

Dielectrics.You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

Energy Storage in Capacitors

Let’s calculate how much work it takes to charge a capacitor.The work required for an external force to move a charge dq through a potential difference V is dW = dq V.

V+ -

+q -q

+dq

From Q=CV ( V = q/C):

qdW V dq dq

C

q is the amount of charge on the capacitor at the time the charge dq is being moved.

We start with zero charge on the capacitor, and end up with Q, so

Q2 2Q Q

0 00

q q QW dW dq .

C 2C 2C

The work required to charge the capacitor is the amount of energy you get back when you discharge the capacitor (because the electric force is conservative).

Thus, the work required to charge the capacitor is equal to the potential energy stored in the capacitor.

2QU .

2C

Because C, Q, and V are related through Q=CV, there are three equivalent ways to write the potential energy.

2 2Q CV QVU .

2C 2 2

2 2Q CV QVU .

2C 2 2

All three equations are valid; use the one most convenient for the problem at hand.

It is no accident that we use the symbol U for the energy stored in a capacitor. It is just another “version” of electrical potential energy. You can use it in your energy conservation equations just like any other form of potential energy!

There are now four parameters you can determine for a capacitor: C, Q, V, and U. If you know any two of them, you can calculate the other two.

Example: a camera flash unit stores energy in a 150 F capacitor at 200 V. How much electric energy can be stored?

2CVU

2

6 2150 10 200U

2

U 3 J

If you keep everything in SI (mks) units, the result is “automatically” in SI units.

Example: compare the amount of energy stored in a capacitor with the amount of energy stored in a battery.A 12 V car battery rated at 100 ampere-hours stores 3.6x105 C of charge and can deliver at least 4.3x106 joules of energy (we’ll learn how to calculate that later in the course).

If a battery stores so much more energy, why use capacitors?Application #1: short pulse magnets at the National Magnet Laboratory, plus a little movie of a short pulse magnet at work.106 joules of energy are stored at high voltage in capacitor banks, and released during a period of a few milliseconds. The enormous current produces incredibly high magnetic fields.

A 100 F capacitor that operates at 12 V can deliver an amount of energy U=CV2/2=7.2x10-3 joules.If you want your

capacitor to store lots of energy, store it at a high voltage.

Truth in advertising: I set up the comparison in a way that makes the capacitor look bad.

Application #2: quarter shrinker.

Application #3: can crusher.

Some links: shrinking, shrinking (can you spot the physics mistake), can crusher,. Don’t do this at home. Or this.