Electric Potential Energy Calculator & Formula

This calculator works out the electric potential energy stored between two point charges. You enter the charge of each object (in coulombs) and the distance between them (in meters), and it returns the energy in joules using the standard physics formula. The result tells you how much work is needed to bring the charges to that separation, or how much energy would be released if they moved apart.

Formula

The electric potential energy U between two point charges is given by Coulomb’s law:

U = k × q� × q₂ / r

• U – electric potential energy in joules (J)
• k – Coulomb’s constant, approximately 8.9875517923 × 10� N·m²/C²
• q� – charge of the first object in coulombs (C)
• q₂ – charge of the second object in coulombs (C)
• r – distance between the centers of the charges in meters (m)

If both charges have the same sign (both positive or both negative), the energy is positive, meaning the system stores energy because the charges repel. If the charges have opposite signs, the energy is negative, indicating an attractive force that would release energy if the charges moved closer together.

Worked Example

Using the default values from the calculator above:

• q� = 2 µC = 2 × 10�� C
• q₂ = −3 µC = −3 × 10�� C
• r = 0.5 m

Plug into the formula:

U = (8.9875517923 × 10�) × (2 × 10��) × (−3 × 10��) / 0.5

First multiply the charges: (2 × 10��) × (−3 × 10��) = −6 × 10�¹² C²

Multiply by k: (8.9875517923 × 10�) × (−6 × 10�¹²) = −5.39253107538 × 10�²

Divide by distance: −5.39253107538 × 10�² / 0.5 = −0.107850621508 J

Rounded to three decimal places, U = −0.108 J. The negative sign shows the charges attract each other.

When to Use This Calculator

This tool is for point charges or objects small enough that their size is negligible compared to the distance between them. Typical uses include:

• Physics homework problems involving Coulomb’s law.
• Estimating the energy stored in a system of charged particles, such as in an atom or a capacitor model.
• Checking the work done to assemble a configuration of charges.

It is not for calculating the energy stored in a capacitor directly (that uses ½CV²) or for continuous charge distributions where you would need integration.

How to Read the Result

The sign of the energy is physically meaningful:

• Positive U – the charges repel; you must do work to bring them closer. The energy equals the work required to move them from infinity to that distance.
• Negative U – the charges attract; energy is released as they come together. The absolute value is the energy needed to separate them completely.

The calculator also gives the force between the charges, which follows the same inverse-square relationship. Force is a vector, but this value is the magnitude along the line connecting them.

Common Mistakes and Unit Pitfalls

Most errors come from unit conversions:

• Microcoulombs (µC) vs coulombs (C): 1 µC = 1 × 10�� C. Forgetting to convert changes the result by a factor of 10¹².
• Distance in cm or mm: Always convert to meters. 1 cm = 0.01 m, 1 mm = 0.001 m.
• Sign of charges: Include the negative sign for negative charges; it affects the sign of the energy.
• Using the wrong constant: Some textbooks use k = 9 × 10� N·m²/C² as an approximation. The calculator uses a more precise value, which is fine for most purposes.

Also, remember that this formula assumes charges are in a vacuum or air. In a material medium, the permittivity changes and the energy would be reduced by the dielectric constant.

Limits of the Formula

The point-charge model breaks down when charges are very close together, especially at the atomic scale where quantum effects dominate. For example, you cannot use this formula to calculate the energy between a proton and an electron at distances smaller than about 10�¹� m without considering quantum mechanics. Also, the formula assumes static charges; moving charges produce magnetic fields and radiation, which are not accounted for here.

Frequently Asked Questions

What is the difference between electric potential energy and electric potential?

Electric potential energy is the energy a charge has due to its position in an electric field, measured in joules. Electric potential (or voltage) is the potential energy per unit charge, measured in volts (joules per coulomb). The potential at a point is independent of the test charge placed there.

Why is the energy negative for opposite charges?

By convention, the potential energy is defined as zero when charges are infinitely far apart. Opposite charges attract, so as they come closer, they lose potential energy (it becomes negative). The more negative the value, the more tightly bound the system is.

Can I use this for a system with more than two charges?

For three or more charges, you calculate the energy for each pair and add them together. The total energy is the sum of all pairwise interactions. This calculator handles only two charges, but you can repeat the calculation for each pair and sum manually.

What if the charges are in water or another material?

In a dielectric medium, the force and energy are reduced by the relative permittivity (dielectric constant) of the material. You would divide the result by the dielectric constant (εr). For water, εr ≈ 80, so the energy would be much smaller than in vacuum.

How do I convert the result to electronvolts (eV)?

1 eV = 1.602 × 10�¹� J. Divide the energy in joules by this number to get electronvolts. This is common in atomic and nuclear physics where energies are very small.

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in q1 | Charge 1 | µC | 2 | 0
in q2 | Charge 2 | µC | -3 | 0
in r | Distance | m | 0.5 | 0
out U | Electric potential energy | J | (8.9875517923e9 * q1 * 1e-6 * q2 * 1e-6) / r | 3
out force | Force between charges | N | (8.9875517923e9 * q1 * 1e-6 * q2 * 1e-6) / (r^2) | 3