Electric Fields Around Charge Configurations

Electric field around two unlike charges

We will start by looking at the electric field around a positive and negative charge placed next to each other. Using the rules for drawing electric field lines, we will sketch the electric field one step at a time. The net resulting field is the sum of the fields from each of the charges. To start off let us sketch the electric fields for each of the charges separately.

A positive test charge (red dots) placed at different positions directly between the two charges would be pushed away (orange force arrows) from the positive charge and pulled towards (blue force arrows) the negative charge in a straight line. The orange and blue force arrows have been drawn slightly offset from the dots for clarity. In reality they would lie on top of each other. Notice that the further from the positive charge, the smaller the repulsive force, \(F_+\) (shorter orange arrows) and the closer to the negative charge the greater the attractive force, \(F_-\) (longer blue arrows).The resultant forces are shown by the red arrows.The electric field line is the black line which is tangential to the resultant forces and is a straight line between the charges pointing from the positive to the negative charge.

Now let's consider a positive test charge placed slightly higher than the line joining the two charges.The test charge will experience a repulsive force (\(F_+\) in orange) from the positive charge and an attractive force (\(F_-\) in blue) due to the negative charge. As before, the magnitude of these forces will depend on the distance of the test charge from each of the charges according to Coulomb's law.Starting at a position closer to the positive charge, the test charge will experience a larger repulsive force due to the positive charge and a weaker attractive force from the negative charge. At a position half-way between the positive and negative charges, the magnitudes of the repulsive and attractive forces are the same. If the test charge is placed closer to the negative charge, then the attractive force will be greater and the repulsive force it experiences due to the more distant positive charge will be weaker. At each point we add the forces due to the positive and negative charges to find the resultant force on the test charge (shown by the red arrows). The resulting electric field line, which is tangential to the resultant force vectors, will be a curve.

Now we can fill in the other field lines quite easily using the same ideas. The electric field lines look like:

Electric field around two like charges (both positive)

For the case of two positive charges \(Q_1\) and \(Q_2\) of the same magnitude, things look a little different. We can't just turn the arrows around the way we did before. In this case the positive test charge is repelled by both charges. The electric fields around each of the charges in isolation looks like.

Now we can look at the resulting electric field when the charges are placed next to each other.Let us start by placing a positive test charge directly between the two charges.We can draw the forces exerted on the test charge due to \(Q_1\) and \(Q_2\) and determine the resultant force.

The force \(F_1\) (in orange) on the test charge (red dot) due to the charge \(Q_1\) is equal in magnitude but opposite in direction to \(F_2\) (in blue) which is the force exerted on the test charge due to \(Q_2\). Therefore they cancel each other out and there is no resultant force. This means that the electric field directly between the charges cancels out in the middle. A test charge placed at this point would not experience a force.

Now let's consider a positive test charge placed close to \(Q_1\) and above the imaginary line joining the centres of the charges. Again we can draw the forces exerted on the test charge due to \(Q_1\) and \(Q_2\) and sum them to find the resultant force (shown in red). This tells us the direction of the electric field line at each point. The electric field line (black line) is tangential to the resultant forces.

If we place a test charge in the same relative positions but below the imaginary line joining the centres of the charges, we can see in the diagram below that the resultant forces are reflections of the forces above. Therefore, the electric field line is just a reflection of the field line above.

Since \(Q_2\) has the same charge as \(Q_1\), the forces at the same relative points close to \(Q_2\) will have the same magnitudes but opposite directions i.e. they are also reflections . We can therefore easily draw the next two field lines as follows:

Working through a number of possible starting points for the testcharge we can show the electric field can be represented by:

Electric field around two like charges (both negative)

We can use the fact that the direction of the force is reversedfor a test charge if you change the sign of the charge that isinfluencing it. If we change to the case where both charges arenegative we get the following result:

Charges of different magnitudes

When the magnitudes are not equal the larger charge will influence the direction of the field lines more than if they were equal. For example, here is a configuration where the positive charge is much larger than the negative charge. You can see that the field lines look more similar to that of an isolated charge at greater distances than in the earlier example. This is because the larger charge gives rise to a stronger field and therefore makes a larger relative contribution to the force on a test charge than the smaller charge.