A2 Physics |
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Capacitance
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
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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.
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.
Non-electrolytic - circuit symbol
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Electrolytic capacitor - circuit symbol
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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).
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”.
Charging a Capacitor Using D.C.
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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: 
The graph of the graph will equal the capacitance of that capacitor.
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Charge Stored in a Capacitor
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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
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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.
From Q=CV. Also the charge can be written as:
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:
Charge, Q, in a capacitor a after it has discharged for t seconds from an initial charge Q0, with resistance, R, and capacitance, C.
The time constant is defined as RC and is measured in seconds to make the whole exponential term dimensionless.
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
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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
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
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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