(e) electric potential energy = Vq = Qq/(4πε₀r) at a distance r from a…

The idea

Electric potential energy of a charge q at distance r from a point charge Q is the work done bringing q from infinity to that point. Since V = Q/(4πε₀r), the energy is E_p = Vq = Qq/(4πε₀r). The sign matters: like charges give positive potential energy that decreases as r increases, while unlike charges give negative potential energy that becomes more negative as r decreases. For Q = +3.0 × 10⁻⁶ C, q = +2.0 × 10⁻⁹ C and r = 0.50 m, E_p = (3.0 × 10⁻⁶ × 2.0 × 10⁻⁹)/(4π × 8.85 × 10⁻¹² × 0.50) ≈ 1.1 × 10⁻⁴ J. Energy is a scalar, so contributions from several charges add algebraically.

What earns the marks

• States that electric potential energy is the work done bringing charge q from infinity to distance r from Q.
• Uses E_p = Vq and substitutes V = Q/(4πε₀r) to obtain E_p = Qq/(4πε₀r).
• Applies the correct sign: like charges give positive E_p, unlike charges give negative E_p.
• Recognises that E_p is a scalar and that contributions from multiple charges add algebraically.
• Calculates E_p for given values of Q, q and r with consistent units.

Common mistakes and how to put them right

• Omitting the sign of the charges.

  Put it right: Like charges give positive potential energy and unlike charges give negative potential energy.

• Confusing electric potential energy with electric potential.

  Put it right: Energy is measured in joules, potential in volts or joules per coulomb.

• Using r measured from the surface of a charged sphere instead of from its centre.

• Treating potential energy as a vector and adding components.

  Put it right: It is a scalar and adds algebraically.

Exam tips

• Write E_p = Qq/(4πε₀r), substitute with signs included, and check the unit is the joule.
• If the question asks for a change in energy, calculate E_p at both separations and subtract.
• Keep ε₀ = 8.85 × 10⁻¹² F m⁻¹ and handle powers of ten carefully to avoid order-of-magnitude errors.