Coulomb's Law Calculator

Charge q1

Charge q2

Separation (r)

Electrostatic force (F)

Relative permittivity of the medium

Enter the two charges and the separation to get the force, or fill any three of the four fields and leave the unknown blank. Keep the permittivity at 1 unless the charges sit in a medium, and switch units on any field.

Please input values into the fields

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Show the force-distance curve

Coulomb's Law is an inverse-square law: move the charges twice as far apart and the force drops to a quarter.

This Coulomb's Law calculator tool allows you to calculate the electrical force between two point charges and also provides a conversion function. By entering in two charges and their distance apart, the electric force can be determined. Alternatively, one of the fields can be left blank for the value to be calculated by the calculator.

The signs of the charges are automatically taken into account so that opposite charges attract and like charges repel. In addition, units such as microcoulombs, centimeters, and newtons can be converted instantly between each other.

What is Coulomb's Law?

Coulomb's law describes the electrostatic force between two stationary charges. Like charges, that is to say two positive or two negative charges, repel each other. Opposite charges, that is to say one positive and one negative charge, attract each other. The larger the product of the two charges, the greater the repulsive or attractive force, and the greater the distance between them, the smaller the electrical force.

The French physicist Charles-Augustin de Coulomb measured this relationship in the 1780s using a precise torsion balance. The law has been named after him since then. It is similar to Newton's Law of Gravity, except that it deals with electricity instead of mass. Both forces decrease as the distance increases, but electrical force can be either attractive or repulsive while gravity is always attractive.

The formula of the Coulomb law:

The electric force between two point charges is the product of the charges multiplied by the Coulomb constant, divided by the square of the distance between them.

F = k\,\frac{q_1 q_2}{r^2}

In this equation F is the electrostatic force, q1 and q2 are two charges, r is the distance between them, and k is Coulomb's constant. This constant determines the strength of the force expressed in everyday units.

k = 8.9875517873681764 \times 10^{9}\ \tfrac{\text{N}\cdot\text{m}^2}{\text{C}^2} \approx 8.99 \times 10^9

If the result is positive then the charges repel each other; if it's negative they attract. This calculator uses this sign convention so that you can immediately tell from the result whether there will be attraction or repulsion.

How to use this calculator:

Pick three known quantities and enter their values. Pick the units for each quantity that are given in the problem statement. Use a minus sign for negative charges. Leave the fourth field blank.

If you have entered three values, the calculation tool will fill in the blank field and display the calculated force as the main result. At the same time, it also displays the stored electrical energy and information about whether the charges attract or repel each other. To calculate a different quantity, delete the known fields and leave another field blank. It is not necessary to select a mode, since the blank field tells the calculation tool which quantity should be calculated.

Calculation of charge or distance.

The Coulomb's law is an equation that contains four variables so each of these can be isolated and solved for. The formula used in the calculator tool is as follows:

q_1 = \frac{F\,r^2}{k\,q_2} \qquad r = \sqrt{\frac{k\,q_1 q_2}{F}}

So the distance between two charges can be calculated if the force is measured and both charges are known. If the force, the distance, and one of the charges are known, then the other charge can be found.

The answer for the above example are calculated as follows:

The transformed form of Coulomb's law:

This table shows the same equation in four different forms, each corresponding to an unknown quantity.

To find	Formula	In words
Force	F = k q1 q2 / r squared	Constant times the product of the charges over the distance squared
Charge q1	q1 = F r squared / (k q2)	Force times distance squared over the constant times the other charge
Charge q2	q2 = F r squared / (k q1)	Force times distance squared over the constant times the other charge
Distance r	r = square root of (k q1 q2 / F)	Square root of the constant times the charges over the force

Example of a calculation:

Two small spheres each have a charge of +1 microcoulomb and +1 microcoulomb respectively and are in air 5 centimeters apart. Since one microcoulomb is equal to one millionth of a coulomb, and 5 centimeters equals 0.05 meters, the force between them is:

F = \frac{(8.99\times10^9)(1\times10^{-6})(1\times10^{-6})}{(0.05)^2} \approx 3.6\ \text{N}

Since both charges are positive, the force of 3.6 newtons manifests as a repulsive force and the two balls push away from each other. If one of the charges were changed to a negative microcoulomb charge, the magnitude would remain 3.6 newtons but the sign would change and the two balls would be attracted to each other.

The Inverse Square Law.

The square of r in the denominator is a hallmark of inverse-square law. If the distance between two charges doubles, then the force does not halve but becomes four times smaller. If the distance triples, the force becomes nine times smaller. The curves for force and distance in the calculators show this steep decrease with respect to charge. This explains why the forces within atoms are extremely large while the forces between everyday objects that are a meter apart are very small.

Gravity also follows the same inverse square law, so the two laws are very similar in form. The difference is in the strength and sign of the constants. The electrostatic attraction between a proton and electron is about 10 to the power of 39 times stronger than the gravitational force between them.

Charges in a medium: The relative permittivity

The basic formula assumes that the charge is in a vacuum, which is approximately equivalent to air. If a charge is brought into another material such as water or glass, the force will be reduced because the medium partially shields the charge. The calculation tools take this into account by using relative permittivity. Relative permittivity, also called dielectric constant, is represented with the Greek letter ε (epsilon) and an index r.

F = \frac{k\,q_1 q_2}{\varepsilon_r\,r^2}

The relative permittivity of vacuum or air is 1, which has no effect on the result and can be ignored. Since water's relative permittivity is about 80, power will be reduced to about one-eightieth of its value in air. Glass has a value between 4 and 7. The Coulomb constant itself derives from the vacuum permittivity, since k equals 1 divided by 4 pi epsilon naught.

Potential energy

In addition to the force, the calculator also reports the potential energy stored in these two charges. The calculation uses the power rather than the square of the distance.

U = \frac{k\,q_1 q_2}{\varepsilon_r\,r}

A positive potential energy means that work must be done to bring like charges together. When the charges are released this energy is given off. A negative potential energy indicates that unlike charges are in a bound state. To separate the charges additional energy must be put in. This is a physical quantity which can be converted into chemical bond energy or battery voltage.

Application of Coulomb's law

The Coulomb's law is the basis for many physics and chemistry concepts. It explains why atoms stick together since there is electrostatic attraction between a positively charged nucleus and negatively charged electrons. It also explains why hair stands up due to static electricity or why a charged balloon sticks to a wall.

Engineers use this law to design capacitors, copiers and inkjet printers in which charged droplets are precisely placed on paper. Chemists use it to analyze ionic bonds and molecular shapes. Where charges meet, the mutual repulsive and attractive forces described by Coulomb's Law follow the inverse square law.

Assumptions and limitations:

This simple form of Coulomb's law is accurate only in a vacuum. If the medium surrounding the two charges has a relative permittivity, εr, then the force between the two charges is given by F = 1 / (4πε0εr) * q1q2 / r2. For higher charged objects, the electric field at different points in space varies, and so the contributions from each small part of the object must be summed, a process similar to that for calculating gravitational fields as described above. If there are more than two charges, the resulting force on a single charge may be calculated by vector addition of the forces from each of the other charges. This principle is known as the superposition principle.

This law assumes that charges are stationary. Moving charges create magnetic effects and emit radiation. These phenomena cannot be explained by Coulomb's Law alone, but it still forms the basis for electrostatics within a broader electromagnetic theory.

Tips for accurate results:

Be aware of powers of ten. Actual charges are usually not whole numbers in coulombs but rather in the micro- or nanocoulomb range. Since the coulomb charge is so large, leave unit conversions to your calculator and avoid mixing units manually such as microcoulombs and meters. Unless the charges are actually in a medium other than air, assume relative permittivity at 1. Also remember that the sign of a force indicates direction: plus means repulsion, minus means attraction.

Frequently asked questions

What is the formula for Coulomb's law?

The formula for Coulomb's law is F = k*q1*q2 / r² where F is the electrostatic force, k is the Coulomb constant, q1 and q2 are the two charges, and r is the distance between them. The Coulomb constant k has a value of approximately 8.99 times 10 to the power of 9, expressed in newton meters squared per coulomb squared. By rearranging this equation you can solve for either one of the charges or the distance.

How do you determine whether a force is attractive or repulsive?

Compare the signs of the two charges. Like charges, meaning both positive or both negative, will repel each other and produce a positive force. Unlike charges, one positive and one negative, will attract each other and produce a negative force. This calculator will read the signs and label your result with an attraction or repulsion sign.

What is the Coulomb's constant?

The Coulomb constant k is equal to 8.9875517873681764 times 10 raised to the power of 9 newtons per meter squared per coulomb squared and it is usually rounded off to 8.99 times 10 raised to the power of 9. It is a constant that determines the magnitude of electrical force and is equal to the value of one divided by the product of four pi and the permittivity of free space.

How to calculate distance or charge instead of power?

Enter the known values and leave the unknown fields blank. To calculate distance, use r = sqrt((k*q1*q2)/F). To calculate a charge, use q1 = (F*r^2)/(k*q2). There is no mode to select since leaving fields blank tells the calculator which value to solve for.

Does the coulomb law change in water or other substances?

Yes. The surrounding materials can shield the charges and reduce the force due to the relative permittivity or dielectric constant. The full formula is F = k q1 q2 / (epsilon_r r squared). Since the relative permittivity for vacuum and air is 1, while it's about 80 for water and between 4 and 7 for glass, a charge experiencing an equal magnitude in water would experience only about one-eightieth of the force in air.

What is the relationship between Coulomb's law and Newton's law of gravitation?

Both follow the inverse square law, with strength decreasing as distance increases. Newtonian gravity depends on the product of two masses and acts only as an attractive force. Coulomb's Law, by contrast, depends on the product of two charges and can produce both attraction and repulsion. The electric force is much stronger; between protons and electrons it is about 10 to the power of 39 times greater than the gravitational force.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Coulomb's law
Definition, history, the constant, the vector form, and the relationship to permittivity and the inverse-square law.
HyperPhysics: Coulomb's Law
A concise reference on the electric force between point charges with worked relationships.
Khan Academy: Coulomb's law and electric force
Plain-language lessons on electric charge, the electrostatic force, and potential energy.