Graphical Representation of Electric Potential (AQA A Level Physics): Revision Note

Graphical representation of electric potential

• An electric field can be described in terms of the variation of electric potential at different points in the field

  • This is known as the potential gradient

• The potential gradient of an electric field is defined as:

The rate of change of electric potential with respect to displacement in the direction of the field

• A graph of potential $V$ against distance $r$ can be drawn for a positive or negative charge $Q$

• This is a graphical representation of the equation:

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

• The gradient of the $V$-$r$ graph at any particular point is equal to the electric field strength $E$ at that point

• This can be written mathematically as:

$E=-\frac{∆V}{∆r}$

• Where:

  • $E$ = electric field strength (V m⁻¹)

  • $∆V$ = potential difference between two points (V)

  • $∆r$ = displacement in the direction of the field (m)

• The negative sign is included to indicate that the direction of the field strength $E$ opposes the direction of increasing potential

• The key features of this graph are:

  • all values of potential are negative for a negative charge

  • all values of potential are positive for a positive charge

  • as $r$ increases, $V$ against $r$ follows a $\frac{1}{r}$ relation for a positive charge and a $-\frac{1}{r}$ relation for a negative charge

  • the gradient of the graph at any particular point is equal to the field strength $E$ at that point

  • the curve is shallower than the corresponding $E$-$r$ graph

Determining potential difference from a field-distance graph

• The potential difference due to a charge can also be determined from the area under a field-distance graph

• A graph of field strength $E$ against distance $r$ can be drawn for a positive or negative charge $Q$

• This is a graphical representation of the equation:

$E=\frac{Q}{4\mathrm{\pi }{\epsilon }_0{r}^2}$

• The area under the $E$-$r$ graph between two points is equal to the potential difference $∆V$ between those points

• The key features of this graph are:

  • all values of field strength are negative for a negative charge

  • all values of field strength are positive for a positive charge

  • as $r$ increases, $E$ against $r$ follows a $\frac{1}{r^2}$ relation (inverse square law)

  • the area under this graph is the change in electric potential $∆V$

  • the curve is steeper than the corresponding $V$-$r$ graph