Work Done on a Charge (AQA A Level Physics): Revision Note

Work done on a charge

• When a charge moves through an electric field, work is done

• The work done in moving a charge $q$ is given by:

$∆W=q∆V$

• Where:

  • $∆W$ = work done (J)

  • $q$ = magnitude of charge moving in the field (C)

  • $∆V$ = potential difference between two points (J C⁻¹)

Electrical potential difference

• Two points at different distances from a charge will have different electric potentials

  • This is because the electric potential increases with distance from a negative charge and decreases with distance from a positive charge

• Therefore, there will be an electric potential difference between the two points equal to:

$∆V=V_f-V_i$

• Where:

  • $V_f$ = final electric potential (J C⁻¹)

  • $V_i$ = initial electric potential (J C⁻¹)

• The potential difference due to a point charge can be written:

$∆V=\frac{Q}{4\mathrm{\pi }{\epsilon }_0}\left(\frac{1}{r_f}-\frac{1}{r_i}\right)$

• Where:

  • $Q$ = magnitude of point charge producing the potential

  • ${\epsilon }_0$ = permittivity of free space (F m⁻¹)

  • $r_f$ = final distance from charge Q (m)

  • $r_i$ = initial distance from charge Q (m)

Electric potential energy

• The electric potential energy of two point charges is given by:

$E_p=\frac{Q_1{Q}_2}{4\mathrm{\pi }{\epsilon }_{0}r}$

• Where:

  • $E_p$ = electric potential energy (J)

  • $Q_1$, $Q_2$ = magnitudes of the charges (C)

  • $r$ = distance between the centres of the two charges (m)

• The work done on a point charge is therefore equal to the change in electric potential energy

  • When $V=0$, then $E_p=0$

• The change in $E_p$, or work done on a point charge $q$ at a distance $r_1$ from the centre of a larger charge $Q$, to a distance of $r_2$ further away can be written as:

$∆E_p=q∆V=\frac{Qq}{4\mathrm{\pi }{\epsilon }_0}\left(\frac{1}{r_2}-\frac{1}{r_1}\right)$

• Where:

  • $Q$ = charge that is producing the electric field (C)

  • $q$ = charge that is moving in the electric field (C)

  • $r_1$ = initial distance of $q$ from the centre of $Q$ (m)

  • $r_2$ = final distance of $q$ from the centre of $Q$ (m)

• Work is done when a positive charge in an electric field moves against the electric field lines or when a negative charge moves with the electric field lines

• A graph of potential energy $E_p$ against distance $r$ can be drawn for two like charges and two opposite charges

• The gradient of the graph at any particular point is equal to the electric force $F$ at that point

Electrostatic equipotential surfaces

• Equipotential lines (2D) and surfaces (3D) join together points that have the same electric potential

• These are always:

  • perpendicular to the electric field lines in both radial and uniform fields

  • represented by dotted lines (unlike field lines, which are solid lines with arrows)

  • an equal distance from the source charge

• Equipotential surfaces can be drawn to represent a fixed electric potential for a number of scenarios, such as:

  • for a point charge

  • for two or more charges

  • between two oppositely charged parallel plates

Equipotential surface for a point charge

• In a radial field, such as around a point charge, the equipotential lines:

  • are concentric circles around the charge

  • become progressively further apart with distance

• If a charged conducting sphere replaced a point charge, the equipotential surface would be the same 

Equipotential surface for multiple charges

• The equipotential surfaces for a dipole (two opposite charges) and for two like charges are shown below:

• An equipotential surface between two opposite charges can be identified by a central line at a potential of 0 V

  • This is the point where the opposing potentials cancel

• An equipotential surface between two like charges can be identified by a region of empty space between them

  • This is the point where the resultant field is zero

Equipotential surface between parallel plates

• In a uniform field, such as between two parallel plates, the equipotential lines are:

  • horizontal straight lines

  • parallel

  • equally spaced

• The spacing between equipotential lines indicates the strength of the electric field

  • This is because they represent potential gradient

• Hence, equally spaced equipotential lines indicate a region of constant electric field strength