Tricks for Solving Coulomb’s Law Problems

If you clicked on this link you know that Coulomb’s law often a bit of a bear to deal with, you have three variables that tend to have exponents, and squaring and a constant, not to mention that if you have more than one force you have to add vectors and that too can be confusing. So, I came up with tricks that often make your life so much easier.  In fact, you can sometimes do the problems in your head without a calculator!

Table of Contents

Trick 1: Combining and separating terms before Calculator work

Trick 2: Change the units of Coulomb’s constant

Trick 3: Draw the force vectors and then ignore the sign of the charge

Trick 4: Adding 2 identical vectors

Trick 1: Combining and separating terms before Calculator work

In Coulomb’s law problems, you often have a lot of numbers or variables to deal with and it is common to type everything into a calculator which increases the calculator errors and makes your life much more difficult.

Example: 40 uC and 30 uC 60 cm apart.

But I can make it much simpler than that, because those units, uC and cm are so common, we can change the units of Coulomb’s constant to simplify it even further.  Which leads to trick #2:

Trick 2: Change the units of Coulomb’s constant

K = 9 x 10^9 Nm^2/C^2 OK, that doesn’t work very well because meters is a big distance and most problems are in cm not meters and even more importantly, Coulombs are ridiculously big and most problems are given in uC, so I thought, let’s just change the units of k.  I will show you how but don’t worry if you are having trouble with the transformation, you just need to use the result. 

So there are 10^2 cm = 1 m and 10^-6 uC = 1^-6 C10^(9-12+4 = 1) = 90

Example 1: 40 uC and 30 uC60 cm apart

Remember, you can’t always use this trick, but, if for example you have two objects 1 meter apart just use 100 cm, or 100 nC = 0.1 uC.  It doesn’t always work but I bet it helps about 60% of the number problems.  You can also use this for Electric fields but remember that the results will be in N/uC not N/C

Example 2: Electric field 30 cm from a 10 uC charge.

However, I would not recommend this trick for Electric potential energy or electric potential, the units will not work out well.  For those two, just use trick #1, separating the units and the exponents. 

Example 3: Potential energy of a 10 uC and a 20 uC charges 30 cm apart

Now let’s go back to Coulomb’s law.  Often students get very confused when there are three or more charges and they are asked to find the total force, especially if any of the charges are negative.  Which leads to trick #3.

Trick 3: Draw the force vectors and then ignore the sign of the charge

Here is the thing, the positive and negative thing with electric charge is arbitrary and all that matters is that opposites attract and like repel.  See, in the mid 1700s Benjamin Franklin (Show me the money) made a machine to generate static electricity that used a brush that rubbed against a glass tube.  Franklin determined that charges weren’t created or destroyed only moved from one object to another. 

Since brooms sweep dirt, Franklin decided that the broom swept up the charges and said that it had positive “electrical fire” and the glass had a lack of positive which he called negative charge.  It took 150 years for JJ Thomson to discover the electron and find that the glass tube was actually gaining negative electrons and leaving the brush with a positive net charge.  Anyway, in Coulomb’s law problems you only need the sign of the charges to determine the direction of force and then you can ignore it. 

Example 1: total force if you add a 20uC between the 40 uC and 30 uC charges 60 cm apart

Example 2: all charges are negative

Example 3: middle charge is negative

What if you have charges that are not in a line?  Well, then you end up adding vectors that are at an angle to each other.  You can solve that by splitting both forces into their x and y components and then add those components and then use Pythagorean to find the total value.  Or, if they are identical forces (which is a common event in these kinds of problems), there is a great trick!   

Trick 4: Adding 2 identical vectors

Start with two vectors that are identical…

Remember, the cos(30) = ½ cos (45) = Root2/2 and cos (60) = root 3/2

Also cos(angle) = a/h

Example 1: isometric triangle

Remember: this only works if both forces have the same values.  However, sometimes, you can add two identical forces and then add a third force

Example 2: square.

You can also use this for Electric field. 

Example 3: Electric field from two charges as a function of x

To review, my tricks are:

  1. Separate all of the constants and the exponents and cancel out what you can BEFORE using calculator
  2. If you can, use k = 90 Ncm^2/uC^2
  3. Draw the force vectors and then ignore the signs of the charges
  4. Two identical vectors add to 2Bcos(q)

Hope this helps.  I usually make videos about the history of science with very little math, for example, I have a video about how and why Coulomb came.

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