Section 3.1: Electric Potential Energy
Electric potential energy is the work done in bringing a charge from infinity to a point in the presence of other charges.
For Two Point Charges:
\[ U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r} \]
where \( r \) is the distance between charges \( q_1 \) and \( q_2 \).
Relationship with Electric Field:
\[ \vec{F} = - \nabla U \]
The force on a charge points in the direction of decreasing potential energy.
Example 1
Compute the electric potential energy of two charges, \( q_1 = 2\,\mu C \) and \( q_2 = -3\,\mu C \), separated by 0.5 m in vacuum.
\[ U = \frac{1}{4\pi\epsilon_0} \frac{(2\times10^{-6})(-3\times10^{-6})}{0.5} \approx -0.108\,\text{J} \]
The negative sign indicates an attractive interaction.
Practice Problems
- Calculate the potential energy of three charges forming a triangle.
- A charge \( q = 5\,\mu C \) is brought near a point charge of \( 10\,\mu C \) at 0.2 m. Find \( U \).
- Explain why electric potential energy is zero at infinity.
- For a dipole, compute the potential energy in a uniform electric field.
- Derive the expression for potential energy of a system of multiple point charges using superposition.