Capacitance

Electric fields can be used to store energy. How?

Any two conductors near each other form a CAPACITOR, we say there is “capacitance in the system”

Capacitance

A capacitor is an energy storage device.

A capacitor can store charge: q is removed from one side of the capacitor and placed onto the other, leaving one side with (-q) and the other side with (+q).

We have to do WORK to do this.

If we connect the two conductors with a wire, the charges flow to make the potential difference between conductors = 0.

When this happens, energy is released

Capacitance

Analogy with air tank:

Capacitance

Charge that can be stored in capacitor is proportional to the potential (voltage) which is pushing the charge into the conductors

Constant of proportionality is “C”, the capacitance

Definition of capacitance:

Capacitance

C is just a function of geometry between two conductors

General method to calculate capacitance:

a) Assume a charge on the conductors

b) Calculate the voltage difference due to the charge

` c) The Ratio Q/V is the capacitance

The unit of capacitance is called the Farad (F)One Farad is equal to one Coulomb per Volt ( F = C / V )

insulator

conductor

Typical capacitor geometriesParallel plates

Concentric cylinders

Concentric spheres

Capacitance for Parallel Plates

To calculate capacitance, need to determine the E field between the plates. We use Gauss’ Law, with one end of our gaussian surface closed inside one plate, and the other closed in the region between the plates

Gauss:

So or

Potential = force/q x distance:

Total charge qon inside of plate

E-field

EAq 0

EdV d

A

Ed

EAVqC 00/

VV

area

A

separationd

Capacitances for simple geometries• Parallel Plate Capacitor

• Cylindrical (nested cylinder) Capacitor

• Spherical (nested sphere) Capacitor

Units of the Farad: named after Michael Faraday

d

AC 0

)/ln(2 0 ab

LC

ab

abC

04

1791 –1867

Faraday had little formal education, and knew little of higher mathematics such as calculus, or even trigonometry, or any but the simplest algebra.

Discovered the electromagnetic induction, diamagnetism, and the laws of electrolysis, invented the use of the field lines

His inventions formed the foundation of electric motor technology, and it was largely due to his efforts that electricity became viable for use in technology.

Energy Stored in CapacitorsEnergy stored in a capacitor = like a stretched spring has potential energy associated with it.

How much work is required to charge a capacitor?

It’s the work to move charges from one plate to another, one at a time.

Moving first electron: almost no potential difference, not a lot of work.

But with more and more electrons: the potential difference V gets bigger and bigger, so more work is required

 

Energy Stored in CapacitorsEach electron has to do qe x V of work to get across. Total work is total charge x average potential difference:

The energy stored in a capacitor is the energy required to charge it:

Electric field energy Another way to think of the energy stored in a in a charged capacitor:Imagine the stored energyis contained in the space between the plates

We can calculate an energy DENSITY (Joules per volume): The volume between the plates is A d. Then the energy density (u) is

But we know and so

This is an important result because it tells us that empty space can contain energy, if there is an electric field in the "empty" space.

If we can get an electric field to travel (or propagate) through empty space we can send or transmit energy! This is achieved with light waves (or microwaves or radio waves or whatever)

Dielectrics

“Dielectric” is another word for insulator, a material that does not allow charges to move easily through it.

Why do we write 0, with a little 0 subscript? Because other materials (water, paper, plastic, even air) have different permittivities = k0. Here, k is called the dielectric constant.

In all of our equations where you see 0, you can substitute 0 when considering some other materials

Notice is larger than 1

The nice thing about this is that we can increase the capacitance of a capacitor by filling the space between conductors with a dielectric:

Some Dielectric ConstantsMaterial

Air 1.00054

Polystyrene 2.6

Paper 3.5

Transformer Oil 4.5

Pyrex 4.7

Ruby Mica 5.4

Porcelain 6.5

Silicon 12

Germanium 16

Ethanol 25

Water (20º C) 80.4

Water (50º C) 78.5

Titania Ceramic 130

Strontium Titanate

310

Cd

AC

0

DielectricsA dielectric in an electric field becomes polarized; this

allows it to reduce the electric field in the gap for the same potential difference.

Dielectrics

Inserting a dielectric into a capacitor while either the voltage or the charge is held constant has the same effect – the ratio of charge to voltage increases.

Combining capacitancesCapacitors in series all have the same charge; the total potential difference is the sum of the potentials across each capacitor.

Note that this gives the inverse of the capacitance.

Combining capacitancesCapacitors in parallel all have the same potential difference; the total charge is the sum of the charge on each.

Combining capacitances

We can picture capacitors in parallel as forming one capacitor with a larger area:

Review of Capacitance:

Capacitance of a parallel-plate capacitor:

Energy stored in a capacitor:

A dielectric is a nonconductor; it increases capacitance.

Dielectric constant: