FHSST Physics/Electrostatics/Electric Fields

Electric Fields

We have learnt that objects that carry charge feel forces from all other charged objects. It is useful to determine what the effect of a charge would be at every point surrounding it. To do this we need some sort of reference.

We know that the force that one charge feels due to another depends on both charges (Q₁ and Q₂). How then can we talk about forces if we only have one charge? The solution to this dilemma is to introduce a test charge. We then determine the force that would be exerted on it if we placed it at a certain location. If we do this for every point surrounding a charge we know what would happen if we put a test charge at any location.

This map of what would happen at any point we call a field map. It is a map of the electric field due to a charge. It tells us how large the force on a test charge would be and in what direction the force would be.

Our map consists of the lines that tell us how the test charge would move if it were placed there.

Test Charge

This is the key to mapping out an electric field. The equation for the force between two electric charges has been shown earlier and is:

${\begin{matrix}F=k{\frac {Q_{1}Q_{2}}{r^{2}}}.\end{matrix}}$

(12.2)

If we want to map the field for Q₁ then we need to know exactly what would happen if we put Q₂ at every point around Q₁. But this obviously depends on the value of Q₂. This is a time when we need to agree on a convention. What should Q₂ be when we make the map? By convention we choose Q₂ = + 1C.

This means that if we want to work out the effects on any other charge we only have to multiply the result for the test charge by the magnitude of the new charge.

The electric field strength is then just the force per unit of charge and has the same magnitude and direction as the force on our test charge but has different units:

${\begin{matrix}E=k{\frac {Q_{1}}{r^{2}}}\end{matrix}}$

(12.3)

The electric field is the force per unit of charge and hence has units of newtons per coulomb [N/C].

So to get the force the electric field exerts we use:

${\begin{matrix}F=EQ\end{matrix}}$

(12.4)

Notice we are just multiplying the electric field magnitude by the magnitude of the charge it is acting on.

What do field maps look like?

The maps depend very much on the charge or charges that the map is being made for. We will start off with the simplest possible case. Take a single positive charge with no other charges around it. First, we will look at what effects it would have on a test charge at a number of points.

Positive Charge Acting on Test Charge

At each point we calculate the force on a test charge, q, and represent this force by a vector.

We can see that at every point the positive test charge, q, would experience a force pushing it away from the charge, Q. This is because both charges are positive and so they repel. Also notice that at points further away the vectors are shorter. That is because the force is smaller if you are further away.

If the charge were negative we would have the following result.

Negative Charge Acting on Test Charge

Notice that it is almost identical to the positive charge case. This is important - the arrows are the same length because the magnitude of the charge is the same and so is the magnitude of the test charge. Thus the magnitude of the force is the same. The arrows point in the opposite direction because the charges now have opposite sign and so the test charge is attracted to the charge.

Now, to make things simpler, we draw continuous lines showing the path that the test charge would travel. This means we don't have to work out the magnitude of the force at many different points.

Electric Field Map due to a Positive Charge

Some important points to remember about electric fields:

• There is an electric field at every point in space surrounding a charge.
• Field lines are merely a representation - they are not real. When we draw them, we just pick convenient places to indicate the field in space.
• Field lines always start at a right-angle (90^o) to the charged object causing the field.
• Field lines never cross!

Combined Charge Distributions

We look at the field of a positive charge and a negative charge placed next to each other. The net resulting field would be the addition of the fields from each of the charges. To start off with let us sketch the field maps for each of the charges as though it were in isolation.

Electric Field of Negative and Positive Charge in Isolation

Notice that a test charge starting off directly between the two would be pushed away from the positive charge and pulled towards the negative charge in a straight line. The path it would follow would be a straight line between the charges.

Now let's consider a test charge starting off a bit higher than directly between the charges. If it starts closer to the positive charge the force it feels from the positive charge is greater, but the negative charge does attract it, so it would move away from the positive charge with a tiny force attracting it towards the negative charge. As it gets further from the positive charge the force from the negative and positive charges change and they are equal in magnitude at equal distances from the charges. After that point the negative charge starts to exert a stronger force on the test charge. This means that the test charge moves towards the negative charge with only a small force away from the positive charge.

Now we can fill in the other lines quite easily using the same ideas. The resulting field map is:

Two Like Charges I: The Positive Case

For the case of two positive charges things look a little different. We can't just turn the arrows around the way we did before. In this case the test charge is repelled by both charges. This tells us that a test charge will never cross half way because the force of repulsion from both charges will be equal in magnitude.

The field directly between the charges cancels out in the middle. The force has equal magnitude and opposite direction. Interesting things happen when we look at test charges that are not on a line directly between the two.

We know that a charge the same distance below the middle will experience a force along a reflected line, because the problem is symmetric (i.e. if we flipped vertically it would look the same). This is also true in the horizontal direction. So we use this fact to easily draw in the next four lines.

Working through a number of possible starting points for the test charge we can show the electric field map to be:

Two Like Charges II: The Negative Case

We can use the fact that the direction of the force is reversed for a test charge if you change the sign of the charge that is influencing it. If we change to the case where both charges are negative we get the following result:

Parallel plates

One very important example of electric fields which is used extensively is the electric field between two charged parallel plates. In this situation the electric field is constant. This is used for many practical purposes and later we will explain how Millikan used it to measure the charge on the electron.

Field Map for Oppositely Charged Parallel Plates

This means that the force that a would feel at any point between the plates would be identical in magnitude and direction. The fields on the edges exhibit fringe effects, i.e. they bulge outwards. This is because a test charge placed here would feel the effects of charges only on one side (either left or right depending on which side it is placed). Test charges placed in the middle experience the effects of charges on both sides so they balance the components in the horizontal direction. This isn't the case on the edges.

The Force on a Test Charge between Oppositely Charged Parallel Plates

What about the Strength of the Electric Field?

When we started making field maps we drew arrows to indicate the strength of the field and the direction. When we moved to lines you might have asked "Did we forget about the field strength?". We did not. Consider the case for a single positive charge again:

Notice that as you move further away from the charge the field lines become more spread out. In field map diagrams the closer field lines are together the stronger the field. This brings us to an interesting case. What is the electric field like if the object that is charged has an irregular shape.