Alexander Fufaev
My name is Alexander FufaeV and here I will explain the following topic:

Coulomb's Law: Electric Force Between Charges

Formula

Formula: Coulomb's Law for Two Point Charges
Coulomb's Electric Force as a Function of the Distance Between the Charges
What do the formula symbols mean?

Force

Unit
Electrostatic force (also called Coulomb force) is the attractive or repulsive electric force between two charges \( q_1 \) and \( q_2 \).

Electric charge

Unit
This charge is the property of the first charge carrier participating in the electrical interaction. Depending on the sign of the charge, the charge carrier repels or attracts other charge carriers. A proton (positive sign) attracts an electron (negative sign).

Electric charge

Unit
This charge is the property of the second charge carrier that participates in the electrical interaction.

Distance

Unit
The distance between the charges \( q_1 \) and \( q_2 \). The larger this distance is, the smaller is the electrostatic force between the charges.

Relative permittivity

Unit
This dimensionless quantity describes the medium in which the two charges are located. If the two charges are in vacuum then \( \varepsilon_{\text r} = 1 \). And, if they are in water, for example, then \( \varepsilon_{\text r} = 80 \). The greater the relative permittivity of the medium, the more this medium weakens the force between the charges.

Vacuum Permittivity

Unit
The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value: $$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$
Explanation
Table of contents
  1. Formula
  2. Basics for understanding Coulomb's law Here you will learn about electric charges.
  3. With what force do charges attract each other? The ingredients of Coulomb's law are explained in detail here.
  4. Is the Electric Force Repulsive or Attractive?
  5. Example: Force between electron and proton Here is a typical example of Coulomb's law.
  6. Coulomb's law in a medium Here you will learn how Coulomb's law changes when two charges are in water, for example.
  7. Exercises with Solutions

Video

Basics for understanding Coulomb's law

Electric positive / negative charge - attraction and repulsion Hover the image!
Interaction between positive and negative charges.

There are two different types of electric charge:

  • positive charge

  • negative charge

Why specifically two types of charges and not one or even three? Because the existing experiments (such as the deflection of charges in electric and magnetic fields) could identify precisely TWO types of charges that differ in their force interaction with each other. Positive charges repel each other with a certain force. Negative charges also repel each other. However, a positive and a negative charge attract each other.

How do charges interact with each other?

Equal charges (++ or --) repel each other. Unequal charges (+- or -+) attract each other.

Electric charge is abbreviated with a small \( q \). And the physical unit of electric charge has been defined as the unit »Coulomb« (\(\text C \)), which can also be written as an ampere-second (\(\text{As} \)):

Remember that knowing the units can help you to derive other units or to check the correctness of your transformations of formulas.

Example of a charge in nature

The electric charge of a negatively charged electron is \( q = -1.602 \cdot 10^{-19} \, \text{C} \). As it is the smallest charge that a free particle can have, this value of charge is also referred to as the elementary charge \( e \). The proton, on the other hand, has a positive elementary charge: \( e = +1.602 \cdot 10^{-19} \, \mathrm{C} \). The sign of the charge will determine whether the electric force between two charges is repulsive or attractive.

With what force do charges attract each other?

You now know that there are two different types of charge that exert a repulsive or attractive electric force \( F_{\text e} \) on each other. You probably have a few questions now: How can I calculate this force? What if the charges are of different sizes? What if the distance between them changes? All these questions can be answered in an experiment. Coulomb's law is a typical physical law that was originally discovered by experiment and not by mathematical derivation.

In general, the experiment for Coulomb's Law proceeds as follows: You electrically charge two metallic spheres so that they carry an electric charge, which you prescribe, for example, using a voltage source. Then, you measure the force between the spheres at a specific distance \( r \) apart. You vary both the distance (that is, \(0.1\, \mathrm{m}, 0.2\, \mathrm{m}, 0.3\, \mathrm{m}\)...) and the charges on one sphere \( q_1 \) and the second sphere \( q_2 \) (that is, \(0.5\, \mathrm{C}, 0.6\, \mathrm{C}, 0.7\, \mathrm{C}\)...). A specific experiment for determining Coulomb's Law is, for instance, the Coulomb's Torsion Balance.

Coulomb's Electric Force as a Function of the Distance Between the Charges
Electric force as a function of the distance between the charges. For small distances, the repulsive force (upper curve) or attractive force (lower curve) is VERY large.

All the measured values you have collected are then examined further in a diagram. In a \(F_{\text e}\)-\(r\)-diagram (that is, force as a function of the charge distance) you find out that the electric force between two charges \(q_1 \) and \(q_2 \), which carry the two spheres, is proportional to \( \frac{1}{r^2} \):

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This means: If you double the distance \(r\), the force \( F_{\text e} \) is reduced by a factor of FOUR!

Insight #1

The greater the distance between two electric charges, the smaller the electric force between them. Accordingly, the repulsion or attraction is weaker.

Then look at how the force changes when you use different charge values. To do this, vary the charge of a sphere. Plot the measured values in a \(F_{\text e}\)-\(q_1\) diagram. The result is a linear relationship:

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If you vary the charge \( q_2 \) of the second sphere, you will of course get the same proportionality:

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Insight #2

The larger the charges, the greater the electric force between them. Accordingly, the repulsion or attraction is stronger.

Distance Between Two Charges and their Electric Field
Two charges (proton and electron) at a distance \( r \), which attract each other. And two protons that repel each other.

If you combine the three experimental relationships 1, 2, and 3, you will already get the following proportionality:

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Only the proportionality constant \( K \) needs to be determined in order to fully establish Coulomb's law:

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By measuring the force between two (known) charges and their distance from each other, you can find the required constant \( K \) by rearranging the equation 5. For example, charge the two spheres so that they have \( q_1 = q_2 = 10^{-4} \, \mathrm{C} \) and place them at a distance \( r = 1 \, \mathrm{m} \) from each other. Then you will measure a force \( F_{\text e} = 89.875 \, \mathrm{N} \) between the two charges. Rearrange 5 with respect to \( K \) and enter the measured value provided here:

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Of course, it doesn't matter which charges and which distance you choose, the result for the proportionality constant will always be the same because it is a constant!

Later on in your physics adventures it will turn out that it makes sense to define the Coulomb constant as follows:

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The value of the vacuum permittivity \( \varepsilon_0 \) can be determined by rearranging Eq. 7 for \( \varepsilon_0 \) and then substituting the value of \(K\) (Eq. 6):

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The unit of the vacuum permittivity is usually given in amperesecond by voltmeter. This can easily be done by converting the units:

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The vacuum permittivity \( \varepsilon_0 \) is encountered nearly everywhere electricity and magnetism are present because this fundamental constant dictates the strength of the electromagnetic interaction between electric charges, shaping the universe as we know it. Why precisely this value 8 is prescribed by nature and not another can only be answered by a "higher power."

The journey ends here, because if you summarize everything you have learned so far, you will come to the following physical relationship, which is called Coulomb's law in honour of the French physicist Charles Augustin de Coulomb, who experimented a lot with charges:

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Is the Electric Force Repulsive or Attractive?

Depending on whether \( q_1 \) or \( q_2 \) is positive or negative in Coulomb's Law 10, the electric force \( F_{\text e} \) has a different sign, resulting in a repulsive or attractive effect between the two charges:

  • \( q_1\) and \(q_2 \) are both positive. Then \( F_{\text e} \) is also positive.

  • \(q_1\) and \(q_2 \) are both negative. Then \( F_{\text e} \) is positive, because "-" times "-" equals "+".

  • \(q_1\) is positive and \(q_2\) is negative (or vice versa). Then \( F_{\text e} \) is negative, because "+" times "-" equals "-".

Insight #3

If the force is positive, the charges repel each other. If the force is negative, the charges attract each other.

Example: Force between electron and proton

According to the simplest atomic model of the hydrogen atom, a negatively charged electron orbits the atomic nucleus, which consists of a single positively charged proton. Both the electron and the proton carry the elementary charge:

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According to the model, the radius \(r\) of the electron orbit is:

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What is the electric force \( F_{\text e} \) that the electron and proton exert on each other at this distance? Do they attract or repel each other?

Use Coulomb's law 10 to find the electric force between the two charges:

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Since the force is negative, the electron and proton attract each other!

Coulomb's law in a medium

Coulomb's law 10 only applies in a vacuum or approximately in air. If you place the two charged spheres in water (not salty water, of course, otherwise there will be a short circuit), you will find that the medium between the charges also plays a crucial role and we have not taken it into account! To take other media into account, the Coulomb law becomes:

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Here, \( \varepsilon_{\text r} \) is the introduced relative permittivity (also called dielectric constant). This dimensionless quantity is used to account for the medium between the charges.

  • If the two charges are in vacuum: \( \varepsilon_{\text r} = 1 \).

  • If the two charges are in the air: \( \varepsilon_{\text r} = 1.0006 \approx 1 \)

  • If the two charges are in lukewarm water: \( \varepsilon_{\text r} \approx 81.1 \).

In the next lesson, we will explore the relationship between electric force, given, for example, by Coulomb's Law, and the electric field.

Exercises with Solutions

Use this formula eBook if you have problems with physics problems.

With the following exercises, you practice manipulating the Coulomb's Law.

Exercise #1: A Charged Particle

What force does a charged particle with a charge of \( 0.001 \, \text{C} \) experience when it is \( 0.2\, \text{m} \) away from a sphere with a charge of \( 0.5 \, \text{C} \)?

Solution to Exercise #1

We need to find out the electric force \(F_{\text e}\) between a charged particle with charge \( q_1 = 0.001 \, \text{C} \) and a sphere with charge \( q_2 = 0.5 \, \text{C} \) at a distance \( r = 0.2\, \text{m} \) from each other.

Here you don't need to rearrange Coulomb's Law, but you can directly substitute the given values. Here, the Coulomb constant \( \frac{1}{4\pi \varepsilon_0} \) corresponds to the value \( 8.987 \cdot 10^9 \, \frac{ \text{Vm} }{ \text{As} } \): 1-1 \begin{align} F_{\text e} &~=~ 8.987 \cdot 10^9 \, \frac{ \text{Vm} }{ \text{As} } ~\cdot~ \frac{0.001 \, \text{C} ~\cdot~ 0.5 \, \text{C}}{(0.2\, \text{m})^2} \\\\ &~=~ 1.1 \cdot 10^8 \, \text{N} \end{align}

So, the particle experiences a very large force of \(1.1 \cdot 10^8 \, \text{N}\).

Exercise #2: The Iron Atom

What is the attraction force between the nucleus of an iron atom with a charge of \( 26 \, e \) and an electron at a distance of \( 10^{-12} \, \text{m} \)?

Solution to Exercise #2

The unknown force is the attractive electric force \( F_{\text e} \) between the nucleus of an iron atom, which carries the charge \( q_1 = 26\,e\), and an electron with charge \(q_2 = e \) at a distance \( r = 10^{-12} \, \text{m}\).

The elementary charge has the value \( e = 1.602 \cdot 10^{-19} \, \text{C}\). Thus, the iron nucleus has the charge: 2 \begin{align} q_1 &~=~ 26 ~\cdot~1.602 \cdot 10^{-19} \, \text{C} \\\\ &~=~ 4.168 \cdot 10^{-18} \, \text{C} \end{align}

With the Coulomb constant, the electric force is: 2-1 \begin{align} F_{\text e} &~=~ 8.987 \cdot 10^9 \, \frac{ \text{Vm} }{ \text{As} } ~\cdot~ \frac{4.168 \cdot 10^{-18} \, \text{C} ~\cdot~ 1.602 \cdot 10^{-19} \, \text{C} }{ \left(10^{-12} \, \text{m}\right)^2 } \\\\ &~=~ 0.006 \, \text{N} \end{align}

So, the nucleus of an iron atom exerts a force of \(0.006 \, \text{N}\) on the observed electron.

Exercise #3: The Atomic Nucleus

With what force do two protons in the atomic nucleus repel each other, assuming they are \( 5\cdot 10^{-15} \, \text{m} \) apart?

Solution to Exercise #3

The unknown force is the electric repulsion force \(F_{\text e}\) between two protons at a distance \( r = 5\cdot 10^{-15} \, \text{m} \). A proton carries the elementary charge \( q_1 = q_2 = 1.602 \cdot 10^{-19} \, \text{C}\).

With the Coulomb constant (see part (a)), the electric force is: 3 \begin{align} F_{\text e} &~=~ 8.987 \cdot 10^9 \, \frac{ \text{Vm} }{ \text{As} } ~\cdot~ \frac{ \left( 1.602 \cdot 10^{-19} \, \text{C} \right)^2 }{ \left( 5\cdot 10^{-15} \, \text{m} \right)^2 } \\\\ &~=~ 9.22 \, \text{N} \end{align}

So, two protons in an atomic nucleus repel each other with a force of \(9.22 \, \text{N}\).

Exercise #4: Charge of the Spheres

Two equally charged spheres repel each other with a force of \( 0.2 \, \text{N} \) when their distance from each other is \( 1\, \text{m} \). What charge do these spheres carry?

Solution to Exercise #4

The task is to find the charge \(q\) of the spheres. Both carry the same charge. That means \(q = q_1 = q_2\). Given is their repulsion force \( F_{\text e} = 0.2 \, \text{N} \) at a distance \( r = 1 \, \text{m} \) from each other. Thus, we need to rearrange Coulomb's Law for the charge \(q\). Bring \(r^2\) to the left side: 4.1 \[ F_{\text e} \, r^2 ~=~ \frac{1}{4\pi \varepsilon_0} \, q^2 \]

Bring \(4\pi \varepsilon_0\) to the left side: 4.2 \[ 4\pi \varepsilon_0 \, F_{\text e} \, r^2 ~=~ q^2 \]

Take the square root on both sides: 4.3 \[ \sqrt{ 4\pi \varepsilon_0 \, F_{\text e} \, r^2 } ~=~ q \]

Substitute the given values: 4.4 \begin{align} q &~=~ \sqrt{ 4\pi \varepsilon_0 \, F_{\text e} \, r^2 } \\\\ q &~=~ \sqrt{ 4\pi ~\cdot~ 8.854 \cdot 10^{-12} \, \frac{ \text{As} }{ \text{Vm} } ~\cdot~ 0.2 \, \text{N} ~\cdot~ \left( 1 \, \text{m} \right)^2 } \\\\ q &~=~ 4.7 \cdot 10^{-6} \, \text{C} \end{align}

The charged spheres repel each other with a force of \(4.7 \,\mu\text{C}\).

Exercise #5: Distance between the Spheres

Two fixed, charged spheres, both with a charge of \( 10^{-6} \, \text{C} \), exert a force of \( 1 \, \text{N} \) on each other. At what distance are they from each other?

Solution to Exercise #5

To do this, rearrange Coulomb's law for the distance \( r \). Bring \( r^2 \) to the left side: 5.1 \[ F_{\text e} \, r^2 ~=~ \frac{q^2}{4\pi \varepsilon_0} \]

Then bring \( F_{\text e} \) to the right side: 5.2 \[ r^2 ~=~ \frac{q^2}{4\pi \varepsilon_0 \, F_{\text e}} \]

Take the square root on both sides: 5.3 \[ r ~=~ \sqrt{ \frac{q^2}{4\pi \varepsilon_0 \, F_{\text e}} } \]

Now plug in the given values: 5.4 \begin{align} r &~=~ \sqrt{ \frac{ \left( 10^{-6} \, \text{C} \right)^2 }{ 4\pi ~\cdot~ 8.854 \cdot 10^{-12} \, \frac{ \text{As} }{ \text{Vm} } ~\cdot~ 1 \, \text{N} } } \\\\ &~=~ 8.99 \cdot 10^{-3} \, \text{m} \end{align}

The two charged spheres are approximately \( 9 \, \text{mm} \) apart from each other.

Exercise #6: Determining the Electric Field Constant

You want to experimentally determine the electric field constant using Coulomb's law. To do this, you charge two spheres equally with \( 2 \cdot 10^{-8} \, \text{C} \), place them \( 0.01\, \text{m} \) apart from each other, and measure a force of \( 0.036\, \text{N} \). What is the value of the electric field constant in this measurement?

Solution to Exercise #6

Given is the charge \( q_1 = q_2 = 2 \cdot 10^{-8} \, \text{C} \), the distance \( r = 0.01\, \text{m} \), and the force \( F_{\text e} = 0.036\, \text{N} \).

Rearrange Coulomb's law for \( \varepsilon_0 \). Multiply both sides by \( \varepsilon_0 \): 6.1 \[ \varepsilon_0\, F_{\text e} ~=~ \frac{1}{4\pi } \, \frac{q^2}{r^2} \]

Multiply both sides by \( \frac{1}{ F_{\text e} } \): 6.2 \[ \varepsilon_0 ~=~ \frac{1}{4\pi F_{\text e}} \, \frac{q^2}{r^2} \]

Plug in the given values: 6.3 \begin{align} \varepsilon_0 &~=~ \frac{1}{4\pi ~\cdot~ 0.036\, \text{N} } ~\cdot~ \frac{ \left(2 \cdot 10^{-8} \, \text{C}\right)^2 }{(0.01\, \text{m})^2} \\\\ &~=~ 8.84 \cdot 10^{-12} \, \frac{ \text{As} }{ \text{Vm} } \end{align}

Exercise #7: Electric Force between Two Sodium Spheres

Two sodium spheres, each with a mass \(M = 0.001 \, \text{kg} \), are positioned at a distance \( r = 1 \, \text{m} \) from each other. One-tenth of the sphere consists of simply positively charged \(\text{Na}^+\) ions (i.e., one valence electron has been removed from a sodium atom).

What is the magnitude of the electric force \(F_{\text e}\) between the two sodium spheres?

Hint: The mass of a sodium atom is: $$ m ~=~ 23 ~\cdot~ 1.67 \cdot 10^{-27} \, \text{kg} $$

Solution to Exercise #7

To solve the problem, use Coulomb's law. The goal is simply to determine the charges \(q_1\) and \(q_2\) of the two sodium spheres. The distance between the spheres is known: \( r = 1 \, \text{m} \).

Since the two sodium spheres are identical in size and especially since they both lack the same number of electrons, they carry the same charge: \( q:= q_1 = q_2 \). Simply denote them as \(q\). Coulomb's law 1 thus becomes: \[ F_{\text e} ~=~ \frac{1}{4\pi \, \varepsilon_0} \, \frac{q^2}{r^2} \]

Now, only one unknown quantity, \(q\), needs to be determined. It is known that one-tenth of all sodium atoms in a sphere lack a valence electron. A valence electron carries the negative charge \( e = -1.602 \cdot 10^{-19} \, \text{C} \), so one-tenth of the sodium atoms are positively charged (they are called \(\text{Na}^+\) ions). Thus, a single \(\text{Na}^+\) ion carries the charge \( e = +1.602 \cdot 10^{-19} \, \text{C} \). The total charge \(q\) of the sphere is simply the number of \(\text{Na}^+\) ions \(N_{\text e}\) multiplied by the charge \(e\): \( q = N_{\text e} \, e\): \[ F_{\text e} ~=~ \frac{1}{4\pi \, \varepsilon_0} \, \frac{\left(N_{\text e} \, e\right)^2}{r^2} \]

However, the only thing known about their number is that it is one-tenth of the total number \(N\) of all sodium atoms in the sphere: \(N_{\text e} = \frac{1}{10} \, N \): 5 \[ F_{\text e} ~=~ \frac{1}{4\pi \, \varepsilon_0} \, \frac{\left( \frac{1}{10} \, N \,e \right)^2}{r^2} \]

To determine how many sodium atoms make up a sphere, the mass of the sphere must be known (it is: \(M = 0.001 \, \text{kg} \)) and the mass of a single sodium atom. The latter can be read from the periodic table of elements: \(m = 23 \cdot 1.67 \cdot 10^{-27} \text{kg} \). The ratio of the masses corresponds to the number of sodium atoms in the sphere: 6 \[ N ~=~ \frac{M}{m} \]

Simply substitute Eq. 6 into Eq. 5: 7 \[ F_{\text e} ~=~ \frac{1}{4\pi \, \varepsilon_0} \, \frac{\left( \frac{1}{10} \, \frac{M}{m} \,e \right)^2}{r^2} \]

Nice, all quantities in 7 are known. Substitute them to find out with what force the two small spheres repel each other: 8 \[ F_{\text e} ~=~ 1.56 \cdot 10^{15} \, \text{N} \]

The calculated force is not a computational error, but actually so unimaginably large. For comparison: The Earth and the Moon attract each other with a force of \( 10^{20} \, \text{N}\).