Electric Potential (Copy)

Definition of Electric Potential

• Electric potential (V) at a point is defined as:
  The work done per unit positive charge in bringing a small test charge from infinity to that point.

  So:
  V = W / q

  Where:
  V = electric potential (V or J/C)
  W = work done (J)
  q = test charge (C)

• Electric potential is a scalar quantity.
• It is a relative quantity: only differences in potential matter, usually with zero potential at infinity.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Electric Potential Due to a Point Charge

• The electric potential at a distance r from a point charge Q is given by:
  V = Q / (4πε₀r)

  Where:
  Q = charge (C)
  r = distance from the charge (m)
  ε₀ = permittivity of free space = 8.85 × 10⁻¹² C² N⁻¹ m⁻²

• V is positive if Q is positive (repulsion — positive work required).
• V is negative if Q is negative (attraction — negative work done).

Electric Potential Energy Between Two Charges

• The electric potential energy of a pair of point charges Q and q separated by a distance r is:
  Eₚ = Qq / (4πε₀r)

  Where:
  Eₚ = potential energy (J)

• This represents the energy stored in the system of the two charges due to their electric interaction.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Electric Field and Potential Gradient

• The electric field strength at a point is equal to the negative gradient of the electric potential at that point.

  Mathematically:
  E = −dV / dr

  • The negative sign shows that the electric field points in the direction of decreasing potential.
  • In a uniform electric field:
    E = ΔV / Δd
• If the potential changes rapidly over a small distance → strong field.
• If potential changes slowly → weak field.

Graphical Interpretation

• A V–r graph for a point charge has a hyperbolic shape.
• For positive Q, the graph starts high near the charge and drops toward zero at infinity.
• For negative Q, the graph is negative and asymptotically approaches zero from below.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Relation Between Potential and Field Lines

• Equipotential lines:
  • Curves along which the potential is constant.
  • Always perpendicular to electric field lines.
  • No work is done when moving a charge along an equipotential surface.
• The closer the equipotential lines, the stronger the electric field.

Units and Concepts Recap

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Worked Example

• A point charge Q = 4.0 × 10⁻⁶ C is located in free space. What is the electric potential 0.25 m away?

  Using:
  V = Q / (4πε₀r)
  V = (4.0 × 10⁻⁶) / (4π × 8.85 × 10⁻¹² × 0.25)
  V ≈ 1.44 × 10⁵ V