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Physics A-Level

A2 Physics

Gravity Fields and Potentials 
Electric Fields and Potentials 
Capacitance 
Magnetic Fields and Induction
Thermal Physics 
Gas Laws 
Further Mechanics 
Nuclear Physics and Radioactivity
Special Topics 


Capacitance

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Contents:

    Types of capacitors
    How does a capacitor work?
    Capacitance
    Charging a capacitor using D.C.
    Charge stored in a capacitor
    Charging and discharging a capacitor
    Exponential discharge - derivation
    Decay curve
    Capacitors in series and parallel


Types of Capacitors



​Capacitors are short term ‘charge-stores’, like an electric spring. They are like batteries, although a battery produces e-s, capacitors just store them. It consists of two metal plates separated by a layer of insulating material called a dielectric.
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There are 2 types of capacitors:

1) Electrolytic capacitors. These hold much more charge and have to be connected with the correct polarity, otherwise they can explode.
2) Non-electrolytic. These hold less charge and can be connected either way round in a circuit.
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Non-electrolytic - circuit symbol
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Electrolytic capacitor - circuit symbol

How Does a Capacitor Work ?

When two conducting plates are connected to a battery electrons move towards one plate. The positive plate loses electrons as well, eventually leave both plates with equal and opposite charge, +Q and -Q. When a capacitor is charged (i.e. electrons are ‘no longer’ moving) we say that the capacitor has charge Q (even though the overall charge is zero).
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Working voltage = The maximum potential difference that can be applied above which the insulation of the dielectric will break down.

Capacitance

Capacitance, C = "the charge, Q, required to cuase unit potential difference, V, in a conductor. It is measured in Farads (normally mF or μF)".

“1 Farad is the capacitance of a conductor, which has potential difference of 1 volt when it carries a charge of 1 coulomb”. 

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Charging a Capacitor Using D.C.

At any time, t, after the switch is closed, the charge, Q, on the capacitor can be calculated using Q=It.

The variable resistor can be altered to keep the current constant (not easy…).

Keep current constant (e.g. ~10μA), calculate the charge and take a reading of the p.d.

Plot a graph of capacitance against voltage and since:
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The graph of the graph will equal the capacitance of that capacitor.
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Charge Stored in a Capacitor

Electrical potential energy is stored when a capacitor is charged.

The energy stored is equal to the work done to force extra charge Δq (i.e. from q to q+Δq) on to the plates and is given by ΔE=vΔq. Where v is the plate potential difference.

Area under the graph is equal to the energy stored = (½bh)
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Charging and Discharging a Capacitor

When discharging the current decreases with the p.d. (like water emptying out of a tube at the bottom of a bucket). This decrease is exponential. Q0, V0 and I0 are all the initial charge, voltage and current through the capacitor at the initial discharge.
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Exponential Discharge - Derivation

Discharge a capacitor, C, through a resistor, R, when the charge decreases from ‘Q’ to ‘Q-ΔQ’ in time Δt, where V is the voltage across the plates.
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From Q=CV. Also the charge can be written as:
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This is negative since Q decreases. Therefore:
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Although you do not need to know this derivation, it is useful to know where the final equation comes from. When Δt→0 then ΔQ/Δt represents the rate of change of charge and is written as dQ/dt. Therefore the equation becomes:
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Charge, Q, in a capacitor a after it has discharged for t seconds from an initial charge Q0, with resistance, R, and capacitance, C.
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The time constant is defined as RC and is measured in seconds to make the whole exponential term dimensionless.
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Similar equations can be derived for the p.d. and current across and through a capacitor at a time t:
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Decay Curve

Q falls to 1/e of its initial  value (i.e. 0.37Q0) in a time equal to the time constant, RC.

When the initial charge is Q0, 
        •After RC seconds = 0.37 x Q0 
        •After 2RC seconds = 0.37^2 x Q0 
        •After nRC seconds = 0.37^n x Q0.

The time taken to halve, T½,  is always the same: 
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Capacitors in Series and Parallel

Series

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Using Kirchoff's law, the voltage across the 2 capacitors adds to equal the voltage of the cell i.e. V=V1+V2. Since V=Q/C for each capacitor and each capacitor has the same amount of charge flowing through it, an equation can be written for the capacitance of a capacitors in series:
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Parallel

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In parallel the current splits. Therefore: I=I1+I2. Since Q=CV and Q=It then It=CV and I=VC/t. Since the voltage is the same for each capacitor in parallel we can simply cancel it out along with the time to get the following equation for capacitors in parallel:
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