Unit 9: Electric Potential

1. Introduction to Electric Potential

Electric potential is a fundamental concept in electrostatics that provides an alternative approach to analyzing electric fields and forces through energy considerations.

Key Concepts:
  • Electric potential is the electric potential energy per unit charge
  • It's a scalar quantity, measured in volts (V)
  • Only potential differences are physically meaningful
  • Work done by electric field equals negative change in potential
V = U/q
where:
V = electric potential (volts)
U = electric potential energy (joules)
q = charge (coulombs)

2. Electric Potential Difference

Electric potential difference (voltage) represents the work per unit charge required to move a charge between two points in an electric field.

ΔV = -∫E⋅dr
ΔV = -(Vf - Vi) = work/charge
Example: Calculate the potential difference between two points in a uniform electric field of 500 N/C separated by 0.2m parallel to the field.
ΔV = -E⋅d = -(500 N/C)(0.2 m) = -100 V

3. Potential Due to Point Charges

The electric potential due to a point charge follows a 1/r relationship, simpler than the 1/r² relationship for electric field.

V = kq/r
where:
k = 8.99 × 10⁹ N⋅m²/C²
q = source charge
r = distance from charge
Electric potential from point charge
Important Points:
  • Potential decreases with distance
  • Can be positive or negative depending on charge
  • Approaches zero at infinity
  • Superposition applies to potentials

4. Superposition of Electric Potential

Unlike electric fields which add as vectors, electric potentials add algebraically because they're scalar quantities.

Vtotal = V₁ + V₂ + V₃ + ...
Vtotal = k(q₁/r₁ + q₂/r₂ + q₃/r₃ + ...)
Example: Find the electric potential at point P due to two charges: +3μC at 2m and -2μC at 3m from P.
V = k(3×10⁻⁶/2 - 2×10⁻⁶/3) V

5. Equipotential Surfaces

Equipotential surfaces are surfaces where the electric potential is constant. They provide valuable insight into the structure of electric fields.

Properties:
  • Electric field lines are perpendicular to equipotential surfaces
  • No work is done moving along equipotential surfaces
  • Closer spacing indicates stronger electric field
  • Conductors are equipotential surfaces in electrostatic equilibrium
Equipotential surfaces diagram

6. Electric Potential Energy

Electric potential energy represents the stored energy of a system of charges due to their positions relative to each other.

U = qV
For two point charges: U = kq₁q₂/r
Example: Calculate the potential energy of two protons separated by 1×10⁻¹⁰ m.
U = k(1.6×10⁻¹⁹)(1.6×10⁻¹⁹)/(1×10⁻¹⁰) = 2.3×10⁻¹⁸ J

7. Relationship Between Field and Potential

Electric field and potential are related through spatial derivatives, providing two complementary ways to analyze electric phenomena.

E = -∇V
In one dimension: E = -dV/dx
Applications:
  • Finding field from potential gradient
  • Finding potential from field integration
  • Analyzing electron motion in potential differences

8. Conductors and Electric Potential

Understanding how conductors behave in terms of electric potential is crucial for many practical applications.

Properties:
  • All points in a conductor at same potential in electrostatic equilibrium
  • Excess charge distributes on surface
  • Field inside conductor is zero
  • Surface is an equipotential

9. Applications and Devices

Electric potential concepts are fundamental to many practical devices and applications.

Common Applications:
  • Capacitors and energy storage
  • Electron microscopes
  • Particle accelerators
  • Voltage sources and batteries
Example: An electron is accelerated through a potential difference of 100V. Calculate its final speed.
½mv² = qΔV
v = √(2qΔV/m)

10. Practice Problems and Review

Problem 1: Three point charges form an equilateral triangle of side length a. Two charges are +q and one is -q. Find the electric potential at the center of the triangle.
Problem 2: A hollow conducting sphere of radius R carries charge Q. Find the electric potential:
a) at the surface
b) at the center
c) at a distance 2R from the center
Problem-Solving Strategy:
  1. Identify known quantities and target variable
  2. Choose appropriate equation(s)
  3. Consider symmetry and boundary conditions
  4. Solve mathematically
  5. Check units and reasonableness