Electric Potential (CIE A Level Physics)

• The electric potential at a point is defined as:

The work done per unit charge in taking a small positive test charge from infinity to a defined point

• Electric potential is measured in J C⁻¹ or V
• It is a scalar quantity but has a positive or negative sign to indicate the sign of the charge
• In a similar way to gravitational potential, electric potential also has a value of zero at infinity

• The electric potential at a point depends on:
  • the magnitude of the point charge
  • the distance between the charge and the point

Electric potential for a positive charge

• Around an isolated positive charge, electric potential: 
  • has a positive value
  • increases when a test charge moves closer
  • decreases when a test charge moves away

For a positive charge, the electric potential decreases in the direction a positive test charge would move in due to the electrostatic repulsion

Electric potential for a negative charge

• Around an isolated negative charge, electric potential: 
  • has a negative value
  • decreases when a test charge moves closer
  • increases when a test charge moves away

For a negative charge, the electric potential decreases in the direction a positive test charge would move in due to the electrostatic attraction

• The electric potential due to a point charge can be calculated using: 

• Where:
  • V = electric potential (V)
  • Q = magnitude of the charge producing the potential (C)
  • r = distance from the centre of the point charge (m)
  • ε₀ = permittivity of free space (F m⁻¹)

• For a positive (+) charge:
  • potential V_e increases as the separation r decreases
  • energy must be supplied to a positive test charge to overcome the repulsive force

• For a negative (−) charge:
  • potential V_e decreases as the separation r increases
  • energy is released as a positive test charge moves in the direction of the attraction force

• The electric potential has an inversely proportional relationship with distance
• Unlike gravitational potential which is always negative, the sign of the charge corresponds to the sign of the electric potential

• Note: this equation also applies to a conducting sphere. The charge on the sphere is treated as if it is concentrated at the centre of the sphere, i.e. like a point charge

Graph of potential against distance for a positive charge

Electric potential is constant inside a charged sphere and decreases with distance outside the sphere

Combining electric potentials

• To find the potential at a point caused by multiple charges, each potential can be combined by addition
• For example, the combined potential of two point charges at a point is:

• Where:
  • Q₁, Q₂ = magnitude of the charges (C)
  • r₁, r₂ = distance between each charge and the point (m)

How to calculate total electric potential

Point X makes an equilateral triangle of length r with two equal positive charges Q. The combined potential of both charges at X is double the potential due to one of the charges