Albert Einstein

Calculating Capacitance

- A capacitor is a system of two insulated conductors.

- The parallel plate capacitor is the
simplest example. When the two conductors have equal but
opposite charge, the
**E**field between the plates can be found by simple application of Gauss's Law.

Assuming the plates are large enough so that theEfield between them is uniform and directed perpendicular, then applying Gauss's Law over surface S_{1}we find,

where A is the area of S_{1}perpendicular to theEfield and σ is the surface charge density on the plate (assumed uniform). Therefore,

everywhere between the plates.

- The potential difference between the plates can be found from

where A and B are points, one on each plate, and we integrate along anEfield line, d is the plate separation, A the plate area and q the total charge on either plate.

- The capacitance (capacity) of this capacitor is defined as,

- The expression for C for all capacitors is the ratio of the magnitude of the total charge (on either plate) to the magnitude of the potential difference between the plates.

- Units of C: Coulomb/Volt = Farad, 1 C/V = 1 F

Note that since the Coulomb is a very large unit of charge the Farad is also a very large unit of capacitance. Typical capacitors in circuits are measured in μF (10^{-6}) or pF (10^{-12}).

- Note that the expression for the capacitance of the parallel plate capacitor depends on the geometric properties (A and d). Even though it appears that there is also a dependence on the charge and potential difference (q/ΔV), what happens is that whatever charge you place on the capacitor the pd adjusts itself so that the ratio q/ΔV remains constant. This is a general rule for all capacitors. The capacitance is set by the construction of the capacitor - not the charge or voltage applied.

- The above expression for the parallel plate capacitor is strictly only true for an infinite parallel plate capacitor - in which "fringing" (see above) does not occur. However, so long as d is small compared to the "size" of the plates, the simple expression above is a good approximation.

- The parallel plate capacitor
provides an easy way to "measure" ε
_{0}

- As indicated above the parallel plate capacitor is the most basic capacitor. You should also be able to determine the expressions for the capacitance of spherical and cylindrical capacitors,

Energy and Capacitors

- One of the most important uses of capacitors is to store electrical energy.

If a capacitor is placed in a circuit with a battery, the potential difference (voltage) of the battery will force electric charge to appear on the plates of the capacitor. The work done by the battery in charging the capacitor is stored as electrical (potential) energy in the capacitor. This energy can be released at a later time to perform work.

- The work necessary to move a charge dq onto one of the plates is given by, dW = Vdq, where V is the pd (voltage) of the battery (= q/C). The total work to place Q on the plate is given by,

which is equal to the stored electrical potential energy, U.

- The electrical energy actually
resides in the electric field
between the plates of the
capacitor. For a parallel
plate capacitor using C =
Aε
_{0}/d and E = Q/Aε_{0}we may write the electrical potential energy,

(Ad) is the volume between the plates, therefore we define the energy density,

- Although we have evaluated this expression for the energy density for a parallel plate capacitor it is actually a general expression. Wherever there is an electric field the energy density is given by the above.

Combinations of Capacitors

It is common to find multiple combinations of capacitors in electrical circuits. In the simplest situations capacitors can be considered to be connected inor inseries.parallel

**Capacitors in Series**

When different capacitors are connected in series the charge on each capacitor is the same but the voltage (pd) across each capacitor is different

In this situation, using the fact that V = V_{1}+ V_{2}+V_{3}we can show that, as far as the voltage source is concerned, the capacitors can be replaced by a single "equivalent" capacitor C_{eq}given by,

**Capacitors in Parallel**

For capacitors connected in parallel it is the voltage which is same for each capacitor, the charge being different.

Using the fact that Q_{Total}= Q_{1}+ Q_{2}+ Q_{3}we can show that the equivalent capacitor, C_{eq}is given by,

At the electric company: *"We would be delighted if you
send in your bill. However, if you don't, you will be."*

*Dr. C. L. Davis*

*Physics Department*

*University of Louisville*

*email*: c.l.davis@louisville.edu