Electric Potential Energy

Electric potential energy ($U$) is the energy stored in a system of charges due to their positions relative to each other. It represents the work needed to assemble the system from infinitely separated charges.

Definition

The electric potential energy of a charge $q$ at a point is the work needed to bring that charge from infinity to that point against the electric force:

$$U = W_{\infty \to point}$$

Potential Energy of Two Point Charges

For two charges $q_1$ and $q_2$ separated by distance $r$:

$$U = \frac{kq_1 q_2}{r}$$

Where:

  • $k = 8.99 \times 10^9$ N·m²/C²
  • Positive $U$: Repulsive (work required to push charges together)
  • Negative $U$: Attractive (work released as charges come together)

Key Properties

Configuration Sign of $U$ Physical Meaning
Like charges ($++, --$) $U > 0$ Repulsive — energy required to bring together
Opposite charges ($+-$) $U < 0$ Attractive — energy released as they approach
$r \to \infty$ $U \to 0$ Zero reference at infinity

Relationship to Electric Potential

$$U = qV$$

Where:

  • $U$ = potential energy of charge $q$
  • $V$ = electric potential at that point

Derivation

Since $V = \frac{U_{test}}{q_{test}}$, then $U = qV$.

Potential Energy of Multiple Charges

For a system of multiple charges, the total potential energy is the sum of potential energies for all pairs:

$$U_{total} = \sum_{all\ pairs} \frac{kq_i q_j}{r_{ij}}$$

Example: Three Charges

$$U = k\left(\frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}}\right)$$

Note: Count each pair only once!

Work and Potential Energy

Work by Electric Field

$$W_{field} = -\Delta U = U_i - U_f$$

Work by External Agent

$$W_{ext} = \Delta U = U_f - U_i$$

Conservation of Energy

$$\Delta K + \Delta U = 0$$

For a charge moving in an electric field: $$\frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = q(V_i - V_f)$$

Examples

Example 1: Potential Energy of Electron and Proton

Find $U$ for electron-proton system at $r = 0.53$ Å (Bohr radius).

$$U = \frac{kq_1 q_2}{r} = \frac{(8.99 \times 10^9)(1.6 \times 10^{-19})(-1.6 \times 10^{-19})}{0.53 \times 10^{-10}}$$

$$U = -4.35 \times 10^{-18} \text{ J} = -27.2 \text{ eV}$$

Example 2: Speed of Accelerated Charge

A proton is accelerated from rest through potential difference $\Delta V = 100$ V. Find its final speed.

$$\Delta K = q\Delta V$$

$$\frac{1}{2}m_p v^2 = e(100)$$

$$v = \sqrt{\frac{2e(100)}{m_p}} = \sqrt{\frac{2(1.6 \times 10^{-19})(100)}{1.67 \times 10^{-27}}}$$

$$v = 1.38 \times 10^5 \text{ m/s}$$

Example 3: Work to Assemble Charges

Find work to bring three charges $+q$ to the vertices of an equilateral triangle of side $a$.

$$U = k\left(\frac{q^2}{a} + \frac{q^2}{a} + \frac{q^2}{a}\right) = \frac{3kq^2}{a}$$

This is the work required (positive for like charges).

Comparison: Potential vs Potential Energy

Aspect Electric Potential ($V$) Electric Potential Energy ($U$)
Definition Property of space Property of charge + field
Depends on Source charges only Source charges AND test charge
Units Volts (V) Joules (J)
Scalar/Vector Scalar Scalar
Relationship $V = U/q$ $U = qV$

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