Revision:Capacitors and capacitance

A capacitor is a device used to store charge.

They can also store energy, in the form of electrical energy due to the electrical field that is generated between the charged plates.

When in a circuit containing a cell and a resistor capacitors collect charge on these plates. One plate collects positive charge and the other collects negative charge. You may therefore say that a capacitor's overall charge is 0 as the plates cancel out. Well not quite as when we talk about the charge a capacitor holds we refer to the amount on just ONE of the plates.

 

 

 

Circuit symbols for capacitors

They are made of two parallel metal plates which are separated by an insulting material called a dielectric. These are two types of main capacitor, electrolytic and non-electrolytic.


Electrolytic capacitors can store a greater amount of charge but are polarised (have a +ve and -ve lead) and therefore must be connected in the correct way. Reversing the polarities of an electrolytic capacitor results in them being damaged and possibly exploding.


Non-electrolytic capacitors store slightly less charge but are not polarised and can be connected anywhere in a circuit.

Capacitance

Definition:

The ability of a capacitor to store charge. It is usually amount of charge in the capacitor divided by the p.d across the capacitor plates

C = \frac {Q}{V}

This can also be written as: Q =CV

Q = charge / C

V = p.d /V

C = Capacitance/ CV^{-1}

The SI unit for capacitance is known as the Farad /F.


Capacitance is usually measured in micro Farads \mu F ie 10^{-6} F

This is because 1F is actually a very large capacitance, much larger than is usually required in an electrical circuit.

Graphs

Energy stored in a capacitor

If we plot a graph with Q /C on the Y axis and V /V on the x axis we get a straight line through the origin. We knew this would be the case before we drew the graph because looking at the equation Q=CV tells us it will be directly proportional as it matches up to the equation of straight line which is y=mx+c.

 

Q=CV

y=mx+c


As we have no constant, c, we know the graph will go through the origin.


The gradient of this graph is capacitance (it is Q over V).

If we reverse the axis, so p.d is on the y axis and charge is on the x axis then the gradient is  \frac {1}{C} and the area under the graph is the amount of energy stored.

This is because a volt is defined as the amount of energy per unit of charge, V=\frac{E}{Q}

So  \frac {E}{Q} \times Q = Energy

We also know the area under the graph is a triangle, the area of a triangle is a 1/2 base x height. so the equations for calculating the amount of energy stored in a capacitor is:

 E = \frac {1}{2}QV

We can also substitute Q with CV so the equation can also be written as  E = \frac {1}{2} CV^2

Charging a Capacitor

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Discharging a Capacitor

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Capacitor discharge

Time constant, \tau

The practical problem with an exponential graph is that the dependent variable never reaches a finite value (maximum or 0). For capacitor circuits we use the time constant to enable a practical measure of the time taken for a capacitor to charge or discharge.

In capacitor circuits we make the time constant equal to 1 for convenience.

 

Basic graph showing discharge of a capacitor. The graph is the same shape for Charge, voltage and negative current against time.

This allows for a simple definition of the time constant to be made and as an easy use of the exponential function.

The time costant

 \tau = RC

R = value of resistor in circuit

C = capacitance of the capacitor


By making the time costant = 1 we can therefore work out what percentage the value will drop to.

Q = Q_0 e^{\frac{-t}{RC}}

 

t=$\tau$=RC = 1


Q=Q_0e{\frac{-1}{1}}

Q=Q_0 \times\frac{1}{e}}

if you put  e^{-1} into your calculator you will get an answer of 0.37 to 2 sig fig.


\therefore Q = 37 \% of Q_0

This means that in the time period of 1 time constant the voltage across a capacitor will drop TO 0.37 of its original value, or 37%.

 

Comments

AQA A A2 Physics Module 4

Unfinished will add graphs and fill in gaps soon.