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Coulomb's Law

Coulomb's law deals with the interaction of charged objects and allows us to quantitatively describe it. It is the first equation that is in fact simply an equation and not a definition like v = d/Dt was when we talked about motion.

The relationship between the charges was found by Coulomb who did a series of experiment in which he measured the deflection of one charge when it was brought close to another charge. He found that when one of the charges is increased so did the force between the charges. If he cut one charge in half the force between them was also cut in half.

Coulomb was not able to measure the amount of charge that he put on any object, just the change of charge. He cut the charge in half or doubled it and looked at the reaction.

The picture to the left shows the device COulomb used - a torsion balance.


To summarize: the force depends on either of the charges and is proportional to either of them and it is inversely proportional to the distance, more accurately, the square of the distance.

In order to turn these proportions, shown at right, into an equation we need a proportionality constant. Here is an example: as kids grow, so do their feet. But while clothes sizes for kids are equal to the age (in years), the shoe sizes are not. A three-year old does not wear size three shoes. As the age increases so does their shoe size, but they are not equal. In order to make them equal one would need proportionality constant. I am not sure whether there is one for shoe size, but you can see how two things that are related can be made equal by using such a constant.

The constant for Coulomb’s law is abbreviated with k and is a very big number.






In our model of charge we now added a layer to the description of how charged objects interact. We found a way that allows us to calculate the interaction, the force, between two charges.


There are two different types of problems associated with the Coulomb’s law. One tests your understanding of the equation and the other your ability to manipulate it and solve for one of the variables.

The first type of problem is pictured to the left. The question reads: What happens to the force between two charges when you double their distance. We know from the experiment that when you increase the distance you will decrease the force, because these two variables are indirectly proportional. Looking at the equation, we can see this as well. When you increase the distance you are increasing the denominator of a fraction. Whenever this happens, the whole fraction has to decrease. The thing that makes this difficult is that the equation does not list the distance but the distance squared. You have to consider the distance twice, so to speak. If you double the distance you therefore decrease the force by a factor of FOUR – “doubling twice”.

If you double each charge, the force will increase and here it is obvious that the force will increase by a factor of four because we have two charges to deal with and have to double the force twice.

The second type of problem involves solving the equation and using numbers. The only difficulty in working with the equation is that the numbers we are using are either very small (the charges) or very big (the constant). Therefore we will use scientific notation to make our lives easier. If you need to review the use of scientific notation, check here.

Consider the following problem:

Two charges one of them 4 mC the other 3 mC are separated by a distance of 0.02 m. Calculate the force between them. The pre-fix  m - micro – stands for 10-6 . Charges are measured in very small amounts. Another thing to note is that in solving the problem, the distance needs to be squared. Scientific notation will make this much easier, not because you don’t have to write all the zeros but because the mathematical operation become easier with scientific notation. In the equation for Coulomb’s Law all the variables all connected by multiplication, so I can move them around as I wish, as long as the distance stays in the denominator. The constant k = 9×109 Nm2 / C2.

I worked out the problem below step by step. I am sure you can skip some of them.



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