Objectives:

• To identify electric potential at a point as the work done in bringing a unit charge from infinity to the point; electric potential is zero at infinity
• Electric potential $V = \frac{Q}{4 \pi \epsilon _{0} r}$ at a distance $r$ from a point charge; changes in electric potential
• To understand what a force-distance graph for a point or spherical charge shows and that work done is the area under the graph
• Electric potential energy $E = Vq = \frac{qQ}{4 \pi \epsilon _{0} r}$ at a distance r from a point charge

Electric potential

Electric potential is denoted with the letter $V$ and is the amount of work required to move a unit positive charge from a point at an infinite distance away to a specific point inside an electric field.

Take the image to the left as an example. The $+Q$ charge is in the centre and is shown to have its own electric field around it.

If another positive point charge $+q$ which is initially an infinite distance away were to be moved to a point P in the electric field of $+Q$, it will need to overcome the Coulomb force of repulsion.

Work done is defined as the product of force and distance moved through that force, $W = Fd$. So in this case a force is repelling the $+q$ charge away from $+Q$ so a force is required to move it through a distance within that field. Electric potential is as you have known it since the start of your AS and is the work done per unit charge $V = \frac{W}{Q}$ , this is still valid in this scenario. So electric potential is similar, but described slightly differently, to potential difference. It should therefore be no surprise that the units of potential difference are Volts, V.

The work done in moving the $+q$ charge through this distance is equal to the stored electric potential energy. It is stored because when/ if released, the electrostatic force (or Coulomb force) will repel them apart again and will continue to do so until they are an infinite distance apart.

Variation of electric potential due to distance

For a positive point charge $Q$ , the magnitude of the electric potential $V$ at a distance $r$ from the charge is given by the following equation:

$V = \frac{Q}{4 \pi \epsilon _{0} r}$

This equation is simple to derive and is done by substituing the Coulomb force equation, $F = \frac{qQ}{4 \pi \epsilon _{0} r^{2}}$ into the equation for work done, $W = Fr$:

$W = \frac{qQ}{4 \pi \epsilon _{0} r^{2}} \times r$

$W = \frac{qQ}{4 \pi \epsilon _{0} r}$

This is the equation for electric potential energy

By substituting this equation into the definition of potential difference, $V = \frac{W}{q}$:

$V = \frac{\frac{qQ}{4 \pi \epsilon _{0} r}}{q}$

$V = \frac{Q}{4 \pi \epsilon _{0} r}$

As a positive unit of charge $q$ travels towards a positive point charge  $Q$, it must do work against the field to overcome the repulsive force, which leads to an increase in potential as $q$ gets closer to  $Q$;

Additionally, by reffing to the electric potential equation $V = \frac{Q}{4 \pi \epsilon _{0} r}$, for a given value of $Q$ , one can see that $V$ is inversely proprotional to $r$ , $V \propto \frac{1}{r}$

If the charge $Q$ is a negative charge, the electric potential will decrease as $q$ gets nearer, therefore the potential energy will be negative;

Force-Distance Graph for a point charge

We have already discussed Coulomb’s law and a graph of how the electrostatic force varies with distance  is as follows;

So, to use the previous example (the diagram to the left), for a charge that to be moved in a radial electric field (non-uniform), the variation of Force $F$ by distance $r$ is shown below by;

The area underneath a force-distance graph is the work done.

Page co-written with Nathan B – Thank you!