10.4 Electric Potential Energy

Describing Electric Potential Energy

Electric potential energy measures the work needed to assemble a system of charged particles. When charges are brought together from infinity, energy is either stored or released depending on whether the charges attract or repel each other.

• Like charges (both positive or both negative) have positive potential energy because work must be done against repulsive forces to bring them together
• Unlike charges (one positive, one negative) have negative potential energy because they naturally attract, releasing energy as they come together
• The reference point is at infinity, where the potential energy is defined as zero

The general equation for electric potential energy between two charged objects is:

$U_{E}=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r}=k \frac{q_{1} q_{2}}{r}$ 🔋

Where:

• $U_E$ = electric potential energy (J)
• $q_1, q_2$ = charges of the two objects (C)
• $r$ = distance between the charges (m)
• $\varepsilon_0$ = permittivity of free space constant ($8.85 \times 10^{-12}$ $\frac{C^2}{N \cdot m^2}$)
• $k$ = Coulomb's constant ($8.99 \times 10^9$ $\frac{N \cdot m^2}{C^2}$)

Multiple Charge Systems

For systems with multiple charges, we must consider all possible pairs of interactions. The total electric potential energy is the sum of the potential energies for each unique pair of charges.

• For three charges ($q_1$, $q_2$, and $q_3$), we calculate:

  1. Potential energy between $q_1$ and $q_2$
  2. Potential energy between $q_1$ and $q_3$
  3. Potential energy between $q_2$ and $q_3$

• The total potential energy is the sum of these three interactions: $U_{total} = k\frac{q_1q_2}{r_{12}} + k\frac{q_1q_3}{r_{13}} + k\frac{q_2q_3}{r_{23}}$

• For four charges, we would have six unique pairs to consider

Conservation of Energy in Electric Systems

Electric potential energy is a form of potential energy that can be converted to other forms of energy:

• When opposite charges move closer together, electric potential energy decreases and may convert to kinetic energy
• When like charges move closer together, work must be done against the repulsive force, increasing the electric potential energy
• The total energy (kinetic + potential) remains constant in isolated systems

Practice Problem 1: Two Point Charges

Solution: First, let's calculate the electric potential energy of the system using the equation: $U_E = k\frac{q_1q_2}{r}$

Given:

• $q_1 = +2.0 \times 10^{-6}$ C
• $q_2 = -5.0 \times 10^{-6}$ C
• $r = 3.0$ m
• $k = 8.99 \times 10^9$ $\frac{N \cdot m^2}{C^2}$

Substituting these values: $U_E = (8.99 \times 10^9)\frac{(2.0 \times 10^{-6})(-5.0 \times 10^{-6})}{3.0}$ $U_E = (8.99 \times 10^9)\frac{-10.0 \times 10^{-12}}{3.0}$ $U_E = -29.97 \times 10^{-3}$ J $U_E = -0.03$ J

The negative sign indicates that the charges attract each other. To separate these charges to infinity would require doing work against this attractive force. The work required equals the negative of the potential energy: Work required = $-U_E = -(-0.03$ J$) = 0.03$ J

Practice Problem 2: Three-Charge System

Solution: We need to find the potential energy for each pair of charges and then add them together.

First, let's calculate the distances between each pair:

• Distance between $q_1$ and $q_2$: $r_{12} = 4.0$ m
• Distance between $q_1$ and $q_3$: $r_{13} = 3.0$ m
• Distance between $q_2$ and $q_3$: $r_{23} = \sqrt{4.0^2 + 3.0^2} = \sqrt{16 + 9} = \sqrt{25} = 5.0$ m

Now calculate the potential energy for each pair:

1. For $q_1$ and $q_2$: $U_{12} = k\frac{q_1q_2}{r_{12}} = (8.99 \times 10^9)\frac{(3.0 \times 10^{-6})(-2.0 \times 10^{-6})}{4.0}$ $U_{12} = -13.49 \times 10^{-3}$ J

2. For $q_1$ and $q_3$: $U_{13} = k\frac{q_1q_3}{r_{13}} = (8.99 \times 10^9)\frac{(3.0 \times 10^{-6})(1.0 \times 10^{-6})}{3.0}$ $U_{13} = 9.0 \times 10^{-3}$ J

3. For $q_2$ and $q_3$: $U_{23} = k\frac{q_2q_3}{r_{23}} = (8.99 \times 10^9)\frac{(-2.0 \times 10^{-6})(1.0 \times 10^{-6})}{5.0}$ $U_{23} = -3.6 \times 10^{-3}$ J

Total electric potential energy: $U_{total} = U_{12} + U_{13} + U_{23}$ $U_{total} = -13.49 \times 10^{-3} + 9.0 \times 10^{-3} + (-3.6 \times 10^{-3})$ $U_{total} = -8.09 \times 10^{-3}$ J $U_{total} = -0.00809$ J