Electric Potential Energy (Cambridge (CIE) A Level Physics): Revision Note

Electric potential energy of two point charges

• 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:

  • ΔW = work done (J)

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

  • ΔV = potential difference between two points (J C⁻¹)

• Work is done when

  • a positive charge in an electric field moves against the field lines

  • a negative charge moves with the field lines

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

  • ε₀ = 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)

  • Q₁, Q₂ = magnitudes of the charges (C)

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

• Similar to electric potential, values of electric potential energy depend on the signs of Q₁ and Q₂

  • By definition, potential V = 0 at infinity, therefore E_p = 0 at infinity

• Like electric potential, the value of electric potential energy can be positive or negative

Change in electric potential energy

• There is a change in electric potential energy when one charge moves away from another

  • This is because work must be done on the field to bring similar charges together or to separate opposite charges

  • Conversely, work is done by the field to separate similar charges or to bring opposite charges together

• When a charge q moves away from a charge Q, the change in electric potential energy is equal to:

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

• Where:

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

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

  • r₁ = initial distance of q from the centre of Q (m)

  • r₂ = final distance of q from the centre of Q (m)

Electric potential energy for moving charges

Work is done when moving a point charge away from another charge

• The change in electric potential energy between two charges is analogous to the change in gravitational potential energy between two masses

Comparing gravitational and electric potential energy

When a small mass is lifted on Earth, there is an increase in gravitational potential energy. This is similar to the increase in electric potential energy when a negative charge moves away from a positive charge