### Table of Contents

- Definition
- Formula
- Electric potential energy of two charges
- Electric potential energy of two electrons
- Calculator
- References

### Definition

Two particles with electric charge, which interact through an electric field, will form a system which will have an **electric potential energy**. For example, if a positively charged particle is brought close to a negatively charged particle, an attraction force will develop between the two charged particles, therefore electric potential energy will be created.

**Electric potential energy** is often called **electrostatic potential energy**.

The electric potential energy is not associated with one charge or the other, it’s the potential energy of the system of two electric charges. If we only have one electric charge present, there will be no potential energy because there will be no attraction/repulsion force created.

### Formula

Two electric charges, interacting with each other due to their electric field, have the electric potential energy defined as [1]:

_{1}· q

_{2}) / r

where:

- U [J] – electric potential energy
- k [N·m
^{2}/C^{2}] – Coulomb constant - q
_{1}[C] – electric charge of the first particle - q
_{2}[C] – mass of second body - r [m] – distance between the electric charges

The **Coulomb constant** also known as the electric force constant, or the electrostatic constant, has a fixed value, equal to:

^{9}N·m

^{2}/C

^{2}

The unit of measurement of **electric potential energy** is **joule** [J].

### Electric potential energy of two charges

A particle with the charge of -5 nanocoulombs is at a distance of 10 centimeters away from another charge of 10 nanocoulombs. Calculate the potential energy of the systems form by these two electric charges.

**Step 1**. Convert the electrical charges from [nC] to [C] by multiplying the [nC] value with 10^{-9}:

_{1}= -5 · 10

^{-9}C

q

_{2}= 10 · 10

^{-9}C

**Step 2**. Convert the distance from [cm] to [m] by dividing the [cm] value to 100:

**Step 3**. Calculate the electric potential energy of the system using equation (1):

^{9}· (-5) · 10

^{-9}· 10 · 10

^{-9}) / 0.1 = – 4.494 · 10

^{-6}J

The resultant electric potential energy is negative, which means that there is an attraction force between the two particles. This is valid since one particle has positive charge and the second particle has negative charge.

### Electric potential energy of two electrons

Two electrons, each with an electric charge of -1.60217662·10^{-19} coulombs, are located at a distance of 1 millimetre from each other. Calculate the electric potential energy of the system of two electrons.

**Step 1**. Convert the distance from [mm] to [m] by dividing the [mm] value to 1000:

**Step 2**. Calculate the electric potential energy of the system:

Since both electrons have the same charge (q_{1} = q_{2} = q), equation (1) will be simplified and written as:

^{2}) / r

where q [C] is the electric charge of the electron.

We can now calculate the electric potential energy of the system formed by the two electrons, by using equation (2):

^{9}· (-1.60217662 · 10

^{-19})

^{2}) / r = 2.30701 · 10

^{-25}J

The resultant electric potential energy is positive, which means that there is a repulsive force between the two electrons. This is valid since both electrons have negative charge.

### Calculator

The electric potential energy calculator allows you to calculate the electric potential energy of two charged particles. You need to enter the charge, distance parameters and choose the desired unit of measurement. Click on CALCULATE to view the results.

The default unit of measurement for energy is **Joule**. If you want the result displayed in another unit, use the dropt down list to choose and click the CALCULATE button again.

### References

[1] David Halliday, Robert Resnick, Jearl Walker, Fundamentals of Physics, 7th edition, John Wiley & Sons, 2004.

[2] Benjamin Crowell, Light and Matter – Physics, 2007.

[3] Raymond A. Serway and John W. Jr. Jewett, Physics for Scientists and Engineers, 6th edition, Brooks/Cole Publishing Co.,2004

[4] Jiansong Li, Jiyun Zhao, and Xiaochun Zhang, A Novel Energy Recovery System Integrating Flywheel and Flow Regeneration for a Hydraulic Excavator Boom System, Energies 2020.

[5] Leo H. Holthuijsen, Waves in oceanic and coastal waters, Cambridge University Press, 2007.

[6] Kira Grogg, Harvesting the Wind: The Physics of Wind Turbines, Carleton College, 2005.