Electric Potential Energy Formula & Examples | Calculating Electrostatic Potential Energy – Video & Lesson Transcript | Study.com

Potential Energy Fields Between Two Charges

Charged particles have an associated electric field. Figure 1 illustrates the electric field lines of a negatively charged particle. If Figure 1 illustrated a positively charged particle instead, the electric field lines would point outwards instead of inwards. An electric field is like a charged cloud surrounding a charged particle, and it is this charged cloud that exerts force on other charged clouds. If two particles have the same charge, then the force between them is repulsive. On the other hand, if two particles have opposite charges, the force between them is attractive. This electric force, because it causes charged particles to be attracted or repulsed by each other, dictates how much work is required to move the charges together, and the amount of work required to move a single charge into a given configuration is the potential energy between two charges. A common method for interpreting the electric potential energy of a system is to use an electrostatic potential map which is a 3D visualization of the force felt between charges.

Figure 1: Equipotential lines of the electric field surrounding a negatively charged particle.

Electric Potential Energy Formula

What is the electric potential energy formula ? The electrostatic potential energy formula, is written as {eq}U_e = k \frac{q_1 q_2}{r} {/eq} where {eq}U_e {/eq} stands for potential energy, r is the distance between the two charges, and k is the Coulomb constant which has a value of {eq}8.99 * 10^9 {/eq} {eq}N m^2/C^2 {/eq}. The Coulomb constant is read as eight point nine nine times ten to the ninth Newton meters squared per Coulomb squared. Electric systems obey the laws of conservation like any other system does, and the mechanical energy of an electric system is always constant. To find the total energy of a charged system, both potential and kinetic energy must be taken into account so that {eq}E_{total} = U_e + KE_e {/eq} where E stands for energy and KE stands for kinetic energy.

Electric Potential Energy Units

There are two common ways to measure the electric potential energy of a system. The common electric potential energy units are volts, V, and electron volts, eV. The base units of volts are Joules, a measure of work, per Coulombs, a measure of electric charge, and the base units of volts can be written as J/C where J stands for Joules and C stands for Coulombs. An electron volt is the amount of energy an electron gains when the electric potential of a system is increased by one volt, and electron volts are a commonly used measure of energy in nuclear and particle physics.

Examples of the Electric Potential Energy Equation

This section explores how to calculate electric potential energy using the electric potential energy equation. The following three electric potential energy examples show how to calculate the electric potential energy for different charge configurations.

Example 1: Calculate the electric potential of the two charges pictured in Figure 2. Charge 1 has a charge of -3C, charge 2 has a charge of +5C, and they are a distance of 3 units apart.

Figure 2: Use this configuration of charges to calculate the electric potential between the charges for Example 1.

1) The equation for electric potential is {eq}U_e = k \frac{q_1 q_2}{r} {/eq}. The first step is to write the knowns.

• {eq}q_1 = -3C {/eq}

• {eq}q_2 = +5C {/eq}

• r = 3

Note that the distance between charges is measured from the center of each charge.

2) Plug the values from 1) and the value for the Coulomb constant into the electric potential energy equation.

{eq}U_e = (8.99*10^9) \frac{-3*5}{3} {/eq}

3) Calculate 2).

{eq}U_e = (8.99*10^9) \frac{-3*5}{3} = -4.50*10^{10} {/eq} V.

This negative electric potential is an attractive force between oppositely charged particles.

Example 2: Use Figure 3 to calculate the electric potential energy felt between a charge of -2C and another charge of -2C that are a distance of 7 units away from each other.

Figure 3: Use this charge configuration to find the electric potential between the two charges for Example 2.

1) The equation for electric potential is {eq}U_e = k \frac{q_1 q_2}{r} {/eq}. The first step is to write the knowns.

• {eq}q_1 = -2C {/eq}

• {eq}q_2 = -2C {/eq}

• r = 7

2) Plug the values from 1) and the value for the Coulomb constant into the electric potential energy equation.

{eq}U_e = (8.99*10^9) \frac{(-2)(-2)}{7} {/eq}

3) Calculate 2).

{eq}U_e = (8.99*10^9) \frac{(-2)(-2)}{7} = 5.14*10^{9} {/eq} V

This positive electric potential is a repulsive force.

Example 3: Using Figure 4, will the three charges stay together or will they fly apart? Charge 1 has a charge of +4C, and it is located at (0, 0). Charge 2 has a charge of -2C, and it is located at (3, 4). Charge 3 has a charge of +1C, and it is located at (6, 0).

Figure 4: Use this charge configuration to find the electric potential energy between the three charges for Example 3.

The charges will stay together if the electric potential energy is attractive, and they will fly apart if the electric potential energy is repulsive. To find the total electric potential energy between the charges, the potential between charge 1 and charge 2, between charge 1 and charge 3, and between charge 2 and charge 3 must be found. This can be written as {eq}U_{etotal} = k \sum \frac{q_i q_n}{r_{in}} {/eq} or {eq}U_{e total} = k( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_2}{r_{23}}) {/eq}.

1) Calculate {eq}U_{e12} = k \frac{q_1 q_2}{r_{12}} {/eq}.

• State the knowns: {eq}q_1 = +4C {/eq} and {eq}q_2 = -2C {/eq}

• Calculate the distance r between charges. Since charges 1 and 2 do not lay on the same plane the distance formula is needed: {eq}r = \sqrt((0-3)^2 +(0-4)^2 ) = 5 {/eq}

• Plug the necessary values into the electric potential energy formula: {eq}U_{e12} = k \frac{-2*4}{5} = k \frac{-8}{5} {/eq}

2) Calculate {eq}U_{e13} = k \frac{q_1 q_3}{r_{13}} {/eq}.

• State the knowns: {eq}q_1 = +4C {/eq} and {eq}q_2 = +1C {/eq}

• Calculate the distance r between charges: Since charges 1 and 3 lay on the same plane r = 6.
• Plug the necessary values into the electric potential energy formula: {eq}U_{e12} = k \frac{1*4}{6} = k \frac{4}{6} {/eq}

3) Calculate {eq}U_{e23} = k \frac{q_2 q_3}{r_{23}} {/eq}.

• State the knowns: {eq}q_2 = +1C {/eq} and {eq}q_3 = -2C {/eq}

• Calculate the distance r between charges. Since charges 2 and 3 do not lay on the same plane the distance formula is needed: {eq}r = \sqrt((6-3)^2 +(0-4)^2 ) = 5 {/eq}

• Plug the necessary values into the electric potential energy formula: {eq}U_{e23} = k \frac{-2*1}{5} = k \frac{-2}{5} {/eq}

4) Add the final answers from 1) – 3): {eq}U_{etotal} = k (\frac{-8}{5} + \frac{4}{6} + \frac{-2}{5}) = k \frac{-4}{3} {/eq}

5) The final step is to substitute in the correct value for k and evaluate: {eq}U_{etotal} = (8.99*10^9) \frac{-4}{3} = -1.20*10^{10} {/eq} V

This negative electric potential energy is an attractive force so the three charges will stay together.

Lesson Summary

Electrostatic potential energy is the potential energy associated with charged particles, and it is a measure of the work that is needed to move a single charge into a given configuration. It is calculated with the following formula: {eq}U_e = k \frac{q_1 q_2}{r} {/eq}. The potential energy between two charges can be visualized using an electrostatic potential map which is a three dimensional heat map of electrostatic potential. What variables are needed in the electrostatic potential energy formula? The electric potential energy formula for two particles is dependent on the charge of each particle and the distance between the particles.

When using the electric potential energy equation to calculate electric potential energy examples make sure to use the correct units. The electric potential energy units are measured either in volts, V, or in electron volts, eV. How to calculate electric potential energy depends on the configuration of the charges, and if the charges are not on the same plane the distance formula needs to be used to find the distance between the charges.