Capacitor Tutorial 1 - Capacitance of a Capacitor




Voltage and Charge

Measuring Capacitance

Energy in a Capacitor
Using Calculus



Capacitors are short term charge-stores, a bit like an electrical spring.  They store energy in the form of an electric field.  A battery stores energy as chemical energy, released when the chemical reaction proceeds.  They are used widely in electronic circuits.  At its simplest the capacitor consists of two metal plates separated by a layer of insulating material called a dielectric. 



The picture shows some different capacitors:



The two components on the right are electrolytic capacitors.  The others are non-electrolytic capacitors.


The symbol for a capacitor is shown below:


There are two types of capacitor, electrolytic and non-electrolytic.  We won’t worry at the moment what these terms mean, other than to say:


If we pump electrons onto the negative plate, electrons are repelled from the negative plate.  Since positives do not move, a positive charge is induced.  The higher the potential difference, the more charge is crowded onto the negative plate and the more electrons repelled from the positive plate.  Therefore charge is stored.  The plates have a certain capacitance.



Capacitance depends on:

In a capacitor, an electric field is formed.  If the electric field is too strong, the dielectric will break down and a spark will jump.   The maximum voltage is known as the working voltage.



Capacitance is defined as:


The charge required to cause unit potential difference in a conductor.


Capacitance is measured in units called farads (F) of which the definition is:


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


So we can write from this definition:


Capacitance (F) = Charge (C)

                                     Voltage (V)


In code, this is written:



[Q - charge in coulombs (C); C – capacitance in farads (F); V - potential difference in volts (V)]  


A 1 farad capacitor is actually a very big capacitor indeed so instead we use microfarads (mF) where 1 mF = 1 × 10-6 F.  Smaller capacitors are measured in nanofarads (nF), 1 × 10-9 F, or picofarads (pF), 1 × 10-12 F.  A working voltage is also given.  If the capacitor exceeds this voltage, the insulating layer will break down and the component shorts out.  The working voltage can be as low as 16 volts, or as high as 1000 V.  


Question 1

Write down what is meant by the following terms: 

  • Dielectric

  • Farad

  • Working voltage



Voltage and Charge

The relationship between voltage and charge is one of direct proportionality:


The gradient of this graph gives the capacitance.


You can verify this for yourself by charging up the capacitor.  You use a voltmeter to measure the voltage, and a charge-meter which measures the charge.  The picture shows one:



The voltage rises as we charge up a capacitor, and falls as the capacitor discharges.  The current falls from a high value as the capacitor charges up, and falls as it discharges.



If we connect a capacitor in series with a bulb:

We can say that a capacitor blocks d.c., but allows a.c. to flow. 


The behaviour of capacitors with AC circuits is discussed in Electricity Tutorial 11.


For information on how capacitors are used in Electronics, you can visit the electronics pages of my sister website HERE.


The capacitor does NOT conduct electricity.  The "flow" of a.c. is due to the charge and discharge of the capacitor.


Question 2 

Why does a capacitor appear to allow ac to flow, but not dc? 


Question 3

What is the charge held by a 470 microfarad capacitor charged to a p.d. of 8.5 V?    



Measuring Capacitance

This circuit can be used to measure the value of a capacitor:




We can use:

to work out the charge going onto the plates.  We also know that:



so we can combine the two relationships to give:




we can now write:


This is an experiment that you may well do in class, although details are not expected for the exam.


Question 4

Why does the circuit only operate on the forward half-cycle?


Question 5

A capacitor is connected to a 12-volt power supply by a reed switch operating at 400 Hz.  The ammeter reads 45 mA.  What is the capacitance of the capacitor?




Energy in a Capacitor

When we charge up a capacitor, we make a certain amount of charge move through a certain voltage.  We are doing a job of work on the charge to build up the electric field in the capacitor.  Thus we can get the capacitor to do a job of useful work.


We know that:

1.      Energy = charge × voltage

2.      Q = CV


This second relationship tells us that the charge – voltage graph is a straight line:

The capacitor is charged with charge Q to a voltage V.  Suppose we discharged the capacitor by a tiny amount of charge, dQ.  The resulting tiny energy loss (dW) can be worked out from the first equation:

dW = V × dQ           


This is the same as the area of the pink rectangle on the graph.


If we discharge the capacitor completely, we can see that:


Energy loss = area of all the little rectangles

 = area of triangle below the graph.





By substitution of Q = CV, we can go on to write:  


Question 6 

What is the energy held by a 50 000 mF capacitor charged to 12.0 V? 



Using Calculus to work out the Energy

We have seen a graphical method to work out the energy. 

The capacitor is charged with charge Q to a voltage V.  Suppose we discharged the capacitor by a tiny amount of charge, dQ.  The resulting tiny energy loss (dW) can be worked out from the first equation:

dW = V × dQ           


In calculus notation we can write, using Q as the integrand (term to be integrated):

Since Q = CV, we can rewrite this and use the power rule of integration to write:


When this job of work is done, the amount of energy transferred is E.  Therefore:



We can also use V as the integrand:


Since V = Q/C, we can write:


When this job of work is done, the amount of energy transferred is E.  Therefore:

Maths Note


Differentiation is about determining the gradient of a graph.


There are a number of rules of differentiation  We will use only two here. 

  • Added constants differentiate to 0;

  • Powers differentiate according to this formula:

  • Multiplied constants are multiplied with the result of the formula that has been differentiated.  Suppose the constant is b:



Let us suppose we have a a straight line graph that follows the general relationship:


y = mx + c

 If we want to differentiate this, we get:


This tells us that the gradient is m.


Maths Window


Integration is the reverse process of differentiation.  The idea is shown in the picture below:


The 2x term is the function.  The function to be integrated is sometimes called the integrand.  Therefore we write this in calculus form as:



The dx term shows that the little strips go along the x-axis.  The integration symbol is a fancy capital letter 'S', which means "summed together".  So we now write:


The C term is a constant.  When we differentiated, the constant that was added to the function had a differentiated value of 0.  Now we are applying the process in reverse, we need to have a definite value for the constant.


Here are some rules for integration:

  • A constant is added to the integrated function.  In some cases this might be zero.  In other cases it has a definite value.

  • Constants that multiply a function are multiplied with the result of the integrated function.  Suppose we have a constant, b:

  • The power rule is shown below.  It does not work with x-1.


  • The integral of x-1 is shown below:


Often you will need to integrate between two points.  You may see an equation like this:


This means you have to work out the value of the integral at p and the value of the integral at q, and then subtract one from the other.  This is shown in the picture below.  The constant, C = 0 for the sake of this argument.