S-Cool Revision Summary

S-Cool Revision Summary

How Capacitors Work

The simplest capacitors are big plates of metal close to each other but not touching. When connected to a potential difference (e.g. a battery), the battery tries to push electrons through the wire away from its negative terminal. Although there isn't a complete circuit, you can imagine that you can shove a few extra electrons onto a big sheet of metal . Let's face it, given the choice between being stuck at a negative terminal or going to a neutral metal plate, electrons will get up and move! So you get a flow of electrons to the plate i.e. you get a current without a complete circuit, but only for a short period of time.

Charging Capacitors

What happens to current as time passes?

Charging and Discharging diagram

As explained above, current falls away as it becomes less attractive for electrons to move to the plate from the cell.

Note: The area under the current-time graph is equal to the amount of charge stored on the plates.

Charge builds up - quickly at first (a lot of electrons arriving each second) and then more slowly. We have already said that potential difference is proportional to charge, so the p.d.-time graph is exactly the same shape as the charge-time graph.

When the capacitor is fully charged, the pd across the plates will equal the emf of the cell charging it.

Discharging Capacitors

Charging and Discharging diagram

Initially there is a large current due to the large potential difference across the plates. The current drops as pd drops.

Time Constant

Capacitors discharge exponentially. That means that their charge falls away in a similar way to radioactive material decay. In radioactivity you have a half-life, in capacitance you have a "time constant".

The rate of removal of charge is proportional to the amount of charge remaining.

Time constant T = RC

(R = Resistance mΩ, C = Capacitance mF)

The charge left on a capacitor and its discharge current both drop to half their initial values in 0.69 x RC seconds.

Energy Stored in a Capacitor

The potential difference across the plates of a capacitor is directly proportional to the charge stored on the plates. This gives a straight line through the origin on a voltage-charge graph. The area under this graph gives the energy stored in a capacitor.

E = 1 QV or 1 Q2
or
1  




CV2 Joules
2
2
C
2
 

Capacitors in Series

In series, capacitors will each have the same amount of charge stored on them because the charge from the first one travels to the second one, and so on.

The total charge stored is the charge that was moved from the cell, which equals the charge that arrived at the first capacitor, which equals the charge that arrived at the second, etc...

The voltage of the circuit is spread out amongst the capacitors (so that each one only gets a portion of the total).

Total capacitance CT is calculated using...

Capacitors in Parallel

Two small capacitors in parallel can be thought of as being the same as one big capacitor.

There is just as much 'plate' on the left hand side for the charge to flow into in both of these diagrams.

So adding capacitors in parallel will increase the space available to store charge and will therefore increase the capacitance of the combination.

Total capacitance CT is calculated using

CT = C1 + C2 + C3

Equations

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Symbols

C = capacitance, F

V = voltage, V

Q = charge, C

Qo = initial charge (at time t = 0), C

t = time, s

R = resistance in circuit connected to the capacitor

W = energy stored in a capacitor, J

τ or RC = the time constant

CT = capacitance of the combination, F

C1= capacitance of capacitor 1, F

C2 = capacitance of capacitor 2, F