Dot Physics

Pre Reqs: vectors, forces

This is typically the first topic in the second semester of introductory physics – the interaction between objects with electric charge. There are 4 fundamental forces that physics typically looks at:

  • Gravity – an interaction between objects with mass – wow, I don’t have a post on the universal law of gravity?
  • Electromagnetic – an interaction between objects with electric charge.
  • Weak Nuclear – an interaction between (let me just say for simplicity) leptons.
  • Strong Nuclear – an interaction between hadrons.

I know those last two are complicated – but I am not going to talk about the strong and weak forces.

Coulomb’s Law

Coulomb’s Law is a model for the forces between two charged particles. Here is the model.

i-c16ee6f1bf213b39c161bb0330c7e09b-2010-03-16_la_te_xi_t_1.jpg

Looks complicated, doesn’t it? But that is the best way to represent that. Typically, books might just show the magnitude of the force, but I think you can handle the vector form. Let me draw a picture to go along with this.

i-8a9e605d90a25b438d388dbf2d773efb-2010-03-16_untitled.jpg

Maybe you see the first problem – which force is this equation for? If the vector r is pointing from q1 to q2, then this gives the force that q1 exerts on charge q2. Note that it is important to include the unit vector r-hat in the equation or you would not have both sides of the equation as vectors. Also, the denominator is the magnitude of this vector squared. If you did not know the vector r, but instead had the location of the two charges (this happens a lot), then:

i-0ad625e2d512896a01c1f97335a4bb7e-2010-03-16_la_te_xi_t_1_1.jpg

Also, in that equation, the 1/4pi-epsilon term is a constant. The charges should be in units of Coulombs, and the distances in meters – this will give a force in Newtons. Here is the value for that constant: (some texts just call this k or something)

i-1096f332146de5f2d2a6e65941ebefab-2010-03-16_la_te_xi_t_1_4.jpg

One final note – in this form, the sign of the charge DOES matter, and this a good thing. If the two charges have opposite signs, the direction of the force will be in the opposite direction as the r-hat vector. If both charges have the same sign, then this force will be pushing the two charges away from each other.

Maybe this seems like an unnecessarily complicated form of Coulomb’s Law, but I think it is the most useful in the long run.

Comparison with gravity

It is useful to make a comparison between the universal law of gravity and the Coulomb force. Here is the gravitational force:

i-3c3305388eefa840d1fafe33ffb55577-2010-03-16_la_te_xi_t_1_2.jpg

This looks similar to the Coulomb equation. The big difference is the negative sign. This is there because gravity is always an attractive force but the mass is always positive.

Where does this come from?

This was a great question I had introductory physics one time. Actually, I think it was something like “how do you derive Coulomb’s Law?” The answer: you don’t. Coulomb’s law is a mathematical model based on experimental data. The basic idea is to get two spheres and put some electric charge on them. Find some way to measure the force between them and see how that compares to the distance. There is a trick here – a uniformly charged sphere looks just like a point charge. Coulomb’s law deals with point charges.

An Example

This is what my students would want – a worked out example. Here is my completely made up example. Suppose I have a 3 nanoCoulomb (nC) charged ball at the origin and a -7 nC charged ball at the location < 0.3, -0.4, 0 > m. What is the force on the negatively charged ball? Here is sketch:

i-64c6d5dbf1434aced7ebee9771591d91-2010-03-16_untitled_1.jpg

The first thing I need is the vector r. This is pretty easy since one of the charges is at the origin. I also need the magnitude of the r vector and the unit vector r-hat.

i-5add5768cf7ad47562b10b2c33370388-2010-03-16_la_te_xi_t_1_3.jpg

Now, I just have to put stuff in. Here is what I get:

i-5489167cd2b4b848f46e8babd86dd967-2010-03-16_la_te_xi_t_1_5.jpg

Not too bad, is it?

Superposition

One last very important thing. What if you have more than two charges? The superposition principle applies to the Coulomb force. Basically, this means that if you have three charges (called A, B, and C), then the force on C will be the vector sum of the force on C due to A and the force on C due to B. Here is a picture.

i-cf4e9429266fe6e4c52e76db84e46399-2010-03-16_untitled_2.jpg

I will do an example with more than two charges in a later post.

Comments

  1. #1 lordaxil
    March 18, 2010

    I disagree that Coulomb’s law is purely an empirical model. It may have been discovered that way but, like every other phenomenon in electromagnetism, it follows from Maxwell’s equations, and can be be derived from applying Gauss’ law to a point charge.

  2. #2 Rhett Allain
    March 18, 2010

    @lordaxil,

    I think this could be like the “which came first, the chicken or the egg”. The important thing is that Coulomb’s law is consistent with maxwell’s equations. Those are also essentially experimentally determined.

  3. #3 vinod arora
    March 18, 2010

    Coulombs law is basically a experimental result.The effect of medium between the two charges is experimental results.If you put this law in cgs system it will be easy ,but to satisfy the basic results in electromagnetism ,It is put in this form in SI system .Its inverse law behavior with gravitaional force is one of the beautiful symmetry which exist in nature

  4. #4 rob
    March 18, 2010

    @Rhett

    you wrote “There are 4 fundamental forces that physics typically looks at:”

    typically?

    have you been hiding a fifth force somewhere?

    :)

  5. #5 Rhett Allain
    March 18, 2010

    @Rob,

    Yes – I am saving that for a special occasion.

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