Electric Potential Energy

The Main Idea

Electric Potential Energy ($U_e$) is the energy stored within a system of charges due to their relative positions. It represents the external work required to assemble a specific configuration of charges from an initial state where they are infinitely far apart. Because the electrostatic force is conservative, this energy depends only on the arrangement of the charges, not the path taken to put them there.

A Mathematical Model

The fundamental equation for the electric potential energy between two point charges, $q_1$ and $q_2$, separated by a distance $r$, is:

[math]\displaystyle{ U_e = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r} }[/math]

For a system containing multiple charges, the total potential energy is the sum of the interaction energies for every unique pair in the system:

[math]\displaystyle{ U_{total} = \sum_{i \lt j} \frac{1}{4\pi\epsilon_0} \frac{q_i q_j}{r_{ij}} }[/math]

Key relationships:

• Work and Energy: $\Delta U_e = -W_{field}$
• Potential Relationship: $U_e = qV$, where $V$ is the Electric Potential.

A Computational Model

In a computational environment like VPython, we calculate the potential energy by iterating through pairs of charges.

# Example snippet for two charges
k = 9e9
q1 = 1e-6
q2 = 2e-6
r = mag(particle2.pos - particle1.pos)
Ue = k * q1 * q2 / r

Examples

Simple

Two protons ($q = 1.6 \times 10^{-19} \text{ C}$) are held $1 \times 10^{-10} \text{ m}$ apart. Calculate the potential energy of the system.

• Solution: $U_e = (9 \times 10^9) \frac{(1.6 \times 10^{-19})^2}{1 \times 10^{-10}} = 2.3 \times 10^{-18} \text{ J}$.

Middling

Calculate the total work required to assemble three $1\mu\text{C}$ charges at the corners of an equilateral triangle with side lengths of $0.5 \text{ m}$.

Difficult

Determine the change in potential energy when an electron is moved from a distance $r_1$ to $r_2$ away from a fixed charged plate with a known surface charge density $\sigma$.

Connectedness

1. Major Connection: Essential for Electrical Engineering in understanding capacitance and energy storage in circuits. 2. Industrial Application: Used in the design of particle accelerators and cathode ray tubes.

History

The concept builds upon Charles-Augustin de Coulomb's 18th-century work on electrostatic forces. It was later refined through the development of potential theory by mathematicians like Carl Friedrich Gauss and Pierre-Simon Laplace.

See also

• Electric Potential
• Conservation of Energy

External links

• PhET: Charges and Fields

References

• Knight, R. D. (2017). Physics for Scientists and Engineers.
• OpenStax University Physics Volume 2