Coulomb's Law (Electric Force): What Is It & Why Is It Important? (W/ Examples)

Like charges repel, and opposite charges attract, but how big is this force of attraction? Just as you have an equation to calculate the force of gravity between two masses, there is also a formula for determining the electric force between two charges.

The SI unit of electric charge is the Coulomb (C) and the fundamental charge carriers are the proton, with charge ​+e​, and the electron, with charge ​-e​, where the elementary charge ​e​ = 1.602
$$

× 10^-19 C. Because of this, an object's charge is sometimes represented as a multiple of ​e​.

Coulomb's law, named after French physicist Charles-Augustin de Coulomb, gives the electric force between two point charges ​_q₁​ and ​q₂​ a separation distance ​r_​ apart as:

\(F = k\frac{q_1q_2}{r^2}\)

Where the constant ​k​ is Coulomb's constant, ​k​ = 8.99 × 10⁹ Nm²/C².

The SI unit for electric force is the Newton (N), just as it is with all forces. The direction of the force vector is toward the other charge (attractive) for opposite charges and away from the other charge (repulsive) if the charges are the same.

Coulomb's law, just like the force of gravity between two masses, is an ​inverse square law​. This means that it decreases as the inverse square of the distance between two charges. In other words, charges that are twice as far apart experience a quarter of the force. But while this charge diminishes with distance, it never goes to zero and so has infinite range.

To find the force on a given charge due to multiple other charges, you use Coulomb's law to determine the force on the charge due to each of the other charges individually, and then you add the vector sum of the forces to get the final result.

​Static electricity:​ Coulomb's law is the reason you get shocked when touching a doorknob after walking across the carpet.

When you rub your feet across the carpet, electrons transfer via friction leaving you with a net charge. All of the excess charges on you repel each other. As your hand reaches for the doorknob, a conductor, that excess charge makes the leap, causing a shock!

​The electric force is much more powerful than gravity:​ While there are many similarities between the electric force and the gravitational force, the electric force has a relative strength of 10³⁶ times that of the gravitational force!

Gravity only seems large to us because the earth that we are stuck to is so large, and most items are electrically neutral, meaning they have the same number of protons and electrons.

​Inside atoms:​ Coulomb's law is also relevant to the interactions between atomic nuclei. Two positively charged nuclei will repel each other due to the coulomb force unless they are close enough that the strong nuclear force (which causes the protons to attract instead but only acts at a very short range) wins out.

This is why high energy is needed in order for nuclei to fuse: The initial repulsive forces have to be overcome. The electrostatic force is also the reason electrons are attracted to atomic nuclei in the first place and is why most items are electrically neutral.

​Polarization:​ A charged object, when brought near the neutral object, causes the electron clouds around the atoms in the neutral object to redistribute themselves. This phenomenon is called ​polarization​.

If the charged object was negatively charged, the electron clouds get pushed to the far side of the atoms, causing the positive charges in the atoms to be slightly closer than the negative charges in the atom. (The opposite occurs if it is a positively charged object that is brought close.)

Coulomb's law tells us that the force of attraction between the negatively charged object and the positive charges in the neutral object will be slightly stronger than the repulsive force between the negatively charged object and the neutral object due to the relative distances between charges.

As a result, even though one object is technically neutral, there will still be attraction. This is why a charged balloon sticks to a neutral wall!

​Example 1:​ A charge of +2​e​ and a charge of -2​e​ are separated by a distance of 0.5 cm. What is the magnitude of the Coulomb force between them?

Using Coulomb's law and being sure to convert cm to m, you get:

\(F = k\frac{q_1q_2}{r^2} = (8.99\times 10^9)\frac{(2\times 1.602\times10^{-19})(-2\times 1.602\times10^{-19})}{0.005^2} = -3.69\times 10^{-23} \text{ N}\)

The negative sign indicates this is an attractive force.

​Example 2:​ Three charges sit at the vertices of an equilateral triangle. At the bottom left vertex is a -4​e​ charge. At the bottom right vertex is a +2​e​ charge, and at the top vertex is a +3​e​ charge. If the sides of the triangle are 0.8 mm, what is the net force on the +3​e​ charge?

To solve, you need to determine the magnitude and direction of the forces due to each charge individually, and then use vector addition to find the final result.

Force between the -4​e​ and +3​e​ charge:

The magnitude of this force is given by:

\(F = k\frac{q_1q_2}{r^2} = (8.99\times 10^9)\frac{(-4\times 1.602\times10^{-19})(3\times 1.602\times10^{-19})}{0.0008^2} = -4.33\times 10^{-21}\text{ N}\)

Since these charges have opposite signs, this is an attractive force, and it points along the left side of the triangle toward the -4​e​ charge.

The force between the +2​e​ and +3​e​ charge:

The magnitude of this force is given by:

\(F = k\frac{q_1q_2}{r^2}=(8.99\times 10^9)\frac{(2\times 1.602\times10^{-19})(3\times 1.602\times10^{-19})}{0.0008^2} = 2.16\times 10^{-21}\text{ N}\)

Since these charges have the same sign, this is a repulsive force and points directly away from the +2​e​ charge.

If you assume a standard coordinate system and break each force vector into components, you get:

\(F_{1x} = -F_1\cos60=-4.33\times 10^{-21}\cos60 = -2.165\times 10^{-21}\text{ N}\
F_{1y} = -F_1\sin60=-4.33\times 10^{-21}\sin60 = -3.75\times 10^{-21}\text{ N}\text{ }\
F_{2x} = -F_2\cos60=-2.16\times 10^{-21}\cos60 = -1.08\times 10^{-21}\text{ N}\
F_{2y} = -F_2\sin60=2.16\times 10^{-21}\sin60 = 1.87\times 10^{-21}\text{ N}\)

Adding ​x​ and ​y​ components gives:

\(F_{netx} = F_{1x} + F_{2x} = -2.165\times 10^{-21} – 1.08\times 10^{-21} = -3.245\times 10^{-21}\text{ N}
\
F_{nety} = F_{1y} + F_{2y} = -3.75\times 10^{-21} + 1.87\times 10^{-21} = -1.88\times 10^{-21}\text{ N}\)

You then use the Pythagorean theorem to find the magnitude of the force:

\(F_{net} = \sqrt{(-3.245\times 10^{-21})^2 +
(-1.88\times 10^{-21})^2} = 3.75\times 10^{-21}\text{ N}\)

And trigonometry gives you the direction:

\(\theta = \tan^{-1}\frac{F_{nety}}{F_{netx}} = \tan^{-1}\frac{(-1.88\times 10^{-21})}{(-3.245\times 10^{-21})} = 30\)

The direction is 30 degrees below the negative ​x​ axis (or 30 degrees below the horizontal to the left.)