# Definitions: voltage, potential, potential difference

## Electrical potential at a point (or just 'potential')

*This is the work done per coulomb of charge moving a very small positive test charge from zero to that point.*

The unit of electrical potential is the volt.

The idea here is that we have a big lump of charge and we want to find out how this big lump will influence some other point in space.

A good way to do this is to put a little positive test charge at that point and measure the force on it. We divide by the charge on our little test charge so we get force per unit test charge. Then the actual size of the test charge doesn't matter. This is called the 'electrostatic field strength' due to the big lump at that point.

Now in fact you can't just make the test charge appear from nowhere, you have to bring it from somewhere. You typically bring it from a long way away, so far away that the big lump has almost no effect.

We call this point infinity and it just means a long way away from the big lump in any direction. So in this case our zero is chosen to be at infinity.

If we imagine the big lump has a positive charge, then it will repel our test charge all the time we bring it closer. So we have to do some work, expend a little energy, as we bring the test charge closer and closer. The energy we expend (per coulomb of test charge) getting from infinity to the point we're interested in is called the 'electrostatic potential' due to the big lump at that point.

For a positive lump, the potential around it is always positive because you have to put work in to get your test charge to any point. For a negative lump, the potential around it is always negative because you have to do work to try and stop the test charge getting to that point as they attract each other.

This is why we have to use conventional current. It's because the definition of potential uses a positive test charge not a negative one. So all of our definitions about energy involve the movement of positive charges.

Now for electrostatics (i.e. charged lumps of stuff) we tend to choose infinity as our zero of potential. But for electric circuits we choose the negative terminal of the power supply as our zero of potential.

So the potential at a point in a circuit is the work done per unit test charge moving a small positive test charge from the negative terminal to that point.

## Voltage at a point

If we think about potential at a point we can say things like. 'All of the positive side of the circuit is at 6 volts' or 'Half way along the bulb filament is at a potential of about 3 volts' or 'the potential of the negative side of the circuit is 0 volts'.

A more informal way of describing potential at a point is just to talk about the voltage at that point. So it's okay to say 'the voltage at this point is about 3 volts'.

You can use a voltmeter to measure the voltage at any given point. You connect one end (the 'negative end') of the voltmeter to the negative terminal of the battery (because that's what you've chosen as your zero of potential) and the other end to the point you're interested in.

You can think of this as just a special case of measuring potential difference.

## Potential difference (p.d.)

*This is the work done per coulomb of charge moving a small positive test charge between two points.*

With potential we always chose one point to be at zero energy, e.g. infinity, or the negative terminal of a battery. With potential difference we can choose any two points. For example one point could be at 9 volts and the other at 4 volts, in which case the potential difference would just be 5 volts.

We tend to talk about potential difference because that's what voltmeters measure. You have to connect both ends of voltmeter because you're comparing the potential at one point with the potential at another.