Coulomb force is the force of either attraction or repulsion between two charged bodies. The force is related to the magnitude and charge on the two bodies and the distance between them by Coulomb’s Law:

where

q_{1} and q_{2} is the amount of charge in Coulombs

r is the distance in meters between the charges

k is the Coulomb’s Law constant = 8.99×10^{9} N·m^{2}/C^{2}

The direction of the force depends on the positive or negative charges on the bodies. If the two charges are identical, the force is a repulsive force. If one is positive and the other negative, the force is an attractive force.

This Coulomb force example problem shows how to use this equation to find the number of electrons transferred between two bodies to generate a set amount of force over a short distance.

**Example Problem:**

Two neutrally charged bodies are separated by 1 cm. Electrons are removed from one body and placed on the second body until a force of 1×10^{-6} N is generated between them. How many electrons were transferred between the bodies?

**Solution:**

First, draw a diagram of the problem.

Define the variables:

F = coulomb force = 1×10^{-6} N

q_{1} = charge on first body

q_{2} = charge on second body

e = charge of a single electron = 1.60×10^{-19} C

k = 8.99×10^{9} N·m^{2}/C^{2}

r = distance between two bodies = 1 cm = 0.01 m

Start with Coulomb’s Law equation.

As an electron is transferred from body 1 to body 2, body 1 becomes positive and body two becomes negative by the charge of one electron. Once the final desired force is reached, n electrons have been transferred.

q_{1} = +ne

q_{2} = -ne

The signs of the charges give the direction of the force, we are more interested in the magnitude of the force. The magnitude of the charges are identical, so we can ignore the negative sign on q_{2}. Since the charges are opposite, the direction of the force is an attractive force. The magnitude of the force equation above can be simplified to:

We want the number of electrons, so solve this equation for n.

Now we can enter the values from the problem into the formula. Remember to use 0.01 m for the 1 cm r value to keep the units consistent.

After crunching the numbers with your calculator, you should get a value of

n = 6.59×10^{8}

**Answer:**

It takes a transfer of 6.59×10^{8} electrons to produce an attractive force of 1×10^{-6} Newtons.

Notice how the units all canceled out in the final step to leave a unitless value for n, which is what we wanted. Checking your units is a great second chance at catching errors in your math. It is a terrible feeling to set up a problem correctly and still get the wrong answer because of unit error. That’s how people lose Mars $125-Million Climate Orbiters!

For another Coulomb force example problem, check out Coulomb’s Law Example Problem.