# Interactive textbook: Electric fields

### In the beginning

Electricity has become so interweaved with our lives that it is difficult to imagine life without it! But human life existed before household electricity.

On the other hand, consider the electric force. It has existed for as long as the universe has existed in its present form. All that humans have done is to use that force to do useful work. Before the invention of the simple cell that could power electric currents around circuits, electricity was simply the name for the reason why charged bodies attract, like gravity is the reason why masses attract.

So... what is charge?

**Charge**

The word **charge** has several different meanings that existed before its use by physicists. They could have made up a new word but at the time they preferred to use words that had some meaning related to the property they were naming - which has often caused problems when it is found that the property is not what they thought it was. This is in some ways true for the term charge. If you are looking after a small child then the child could be called your charge, and wherever you went your charge would follow. In the same way, electrical charge seemed to be attached to matter. We now think of it differently.

Charge isn't attached to matter. It's a property of matter itself - just like mass.

**Positive and negative**

It would be difficult to say when mass was discovered since the effects related to it are so obvious. Large masses are difficult to move (as even animals well know). The consequences of charge are not so obvious; experiments have to be done. Ancient Greek philosophers noticed that amber attracted small fibres when rubbed, but it wasn't until the 18th century when Benjamin Franklin did his experiments with kites that the present day model of positive and negative charge began to take shape.

**Charges and the atom**

If matter contains charge and matter is made of atoms, it seems likely that atoms contain charge. In fact, atoms are made of smaller particles called electrons, protons and neutrons and it is these particles that have charge:

- Electrons are negative
- Protons are positive
- Neutrons are (unsurprisingly) neutral

These particles are arranged with the heavy protons and neutrons in a central nucleus and the much lighter electrons around the outside. Below is a representation of an atom drawn to scale. Zoom in on the middle to see the proton and neutron.

Charge is measured in Coulombs; the charge of an electron (\(-e\)) is \(-1.6\times10^{-19}\text{ C} \) and the charge on a proton (\(+e\)) is \(+1.6\times10^{-19}\text{ C} \) so the atom above has zero charge. '\(e\)' is the smallest possible charge and is called the fundamental charge. All charges are multiples of \(e\).

The electron has about \(1\over2000 \)^{th} of the mass of the proton and neutron so is much easier to move around. A negatively charged body has extra electrons and a positively charged body has some electrons missing. Electrons can be moved by rubbing two different materials together. Different materials have different electron affinities; the material with the highest affinity will end up with the most electrons (e.g. rubber has a higher electron affinity than wool).

When we hold two oppositely charged bodies close to each other they attract.

In the last animation the force on the green balloon is:

Newton's 3rd law

In the last animation, if two electrons were removed from the green balloon the force on the yellow balloon would be:

The green balloon would have a charge of +e instead of -e.

### Electric fields

An **electric field** is a region of space where a small positive test charge experiences a force due to its charge.

#### Electric force

**Coulomb's law** states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of their distance apart.

\(F\propto{Q_1Q_2\over r^2}\)

\(F=k{Q_1Q_2\over r^2}\)

- The constant of proportionality is \(k = 9 \times 10^9\text{ N m}^2\text{ C}^{-2}\).

Note that, although the law states point charges, it is also true for spheres of charge. They act as though there was the same charge in their centre so the distance apart is the distance between their centres.

**Electric field strength**

Electric field strength (\(E\)) is defined as the force per unit charge experienced by a small positive test charge (\(q\)) placed at the point in the field.

\(E={F\over q}\)

The field strength a distance \(x \) from a point charge \(Q\) is therefore:

\(E=k{Q\over x^2}\)

Like force, field strength is a vector quantity. Its direction is the same as the force. The unit of field strength is \(\text{N C}^{-1}\).

The simulation below shows how the field strength changes as a test charge is moved around a charged hollow sphere.

Note that the field inside is zero. This is because the charges all pull in different directions so the forces cancel each other out.

**Field lines**

The field around a charged sphere changes in magnitude and direction as the test charge is moved around. To visualise this we draw field lines to show the direction. The magnitude is proportional to the spacing of the lines, with closer lines indicating greater field strength.

The direction of the field lines is that in which a positive charge would be accelerated.

**Radial** fields are produced by point charges:

A **uniform** field can be created between two parallel plates:

We can see the field is uniform because the field lines are parallel and equally spaced. However, edge effects are revealed where the field lines no longer follow this rule.

Calculate the field strength 0.5 m from the surface of a 1m diameter sphere of charged to 1 nC.

9 x 10^{9} x 1 x 10^{-9}/1

An electron experiences a force of 1.6 x 10^{-20} N in an electric field.

Calculate the field strength.

E = F/q = 1.6 x 10^{-20}/1.6 x 10^{-19}

#### Electric potential energy

Potential energy is defined as the work done in taking a body from infinity to a point. Well, if you took a positive charge from infinity and placed it close to another positive charge you would have to do work so it would have potential energy. **Electrical potential energy** (\(E_\text{p}\)) is the work done in taking a charged body from infinity to a point in an electric field.

If you actually did this the force required to move the charge would increase as you got closer to charge creating the field. Therefore, you can't simply use force multiplied by distance. Instead, we use the area under the force-distance graph:

The line has the equation \(F = kQq{1\over r^2}\) . The area can be found by integrating this function with respect to \(r\), giving the solution \(kQq\over r\).

\(E_\text{p}=k{Qq\over r}\)

The work done (\(W\)) moving a charge in a uniform field is much simpler since the force is constant everywhere.

To move the charge, a force would have to be applied that is equal and opposite to the electric force.

\(F=Eq\)

\(W=Fd=Eqd\)

If we were to take the negative plate as our zero in potential energy then the potential energy of a charge \(q\) positioned next to the positive plate is given by:

\(E_\text{p}=Eqd\)

#### Electric potential

**Electric potential** (\(V\)) is defined as the work done **per unit charge** in taking a small positive test charge from infinity to the point in question.

This is the same as the ratio of electric potential energy to charge:

\(V={E_\text{p}\over q}\)

Potential is a scalar quantity (since it is linked to energy and not force). Its unit is \(\text{J C}^{-1}=V\) (the volt).

For the field around a point charge the potential at distance \(r\) is:

\(V=k{Qq\over r}\div q=k{Q\over r}\)

For the example of a uniform field at a distance \(d\) from the zero potential plate:

\(V={Eqd\over q}=Ed\)

You will see later that this is quite a useful result. It enables us to calculate the field strength between two plates if you know the potential difference between them and their separation:

\(E={V\over d}\)

It also shines light on an interesting relationship between field strength and potential. If we draw a graph of the change in potential as we crossed from one plate to the other we would get a straight line:

The gradient of this line is the field strength and this is true for **all** electric fields (radial or uniform):

\(E={\Delta V\over \Delta x}\)

**Potential hills and wells**

If we look at the graph of potential vs position we can not only easily visualise the potential but, by looking at the gradient, we can also see the strength of the field. Below is an example of a potential vs position graph for the potential around two charged spheres.

If this was a graph of gravitational potential on the surface of the earth then the changes in potential would be related to changes in height. We would be looking at two hills separated by a valley or well. We use the same terms in electric fields. A ball placed on the slope would experience a resultant force causing it to roll down the hill. The size of that resultant force is related to the steepness of the slope. Force is related to field strength, so we can see that the resultant force is greatest near the top of the right hand hill. Resultant force is zero at the top of the hills (inside the sphere of charge) and at the bottom of the well.

If the potential difference between two horizontal plates 2cm apart is 10V what is the field strength between the plates?

E = V/d = 10/2 x 10^{-2}

Two horizontal plates 20cm are set up so that the bottom one is at 0V and the top one 10 V where will the potential be 1V?

potential gradient = 0.5 V cm^{-1}

The graph represents a potential well.

The field strength is maximum and zero at:

E = potential gradient

The graph represents a potential well.

The potential is maximum at:

The y axis is potential