Question

"An electron and a proton are attracted to each other by the Coulomb force, similar to the gravitational force. Assume that the electron is in a circular orbit around the proton. Determine the radius of the orbit, if the period is:
a) 24h
b)4x10⁻¹⁶, just like in the hydrogen atom"

Coulomb's force is:

F=8,988x10⁹x((q1xq2)xr²)

and the gravitational force is:

F=6,674x10⁻¹¹x((Mxm)/r²)

the perimeter of a circle is 2 x pi x r.

What am I missing?
Thanks in advance!

Answer 1

Let's see:

This is a question about circular motion and centripetal acceleration. The centripetal acceleration is caused by the centripetal force (a force directed toward the center that causes a body to follow a curved path).

The gist of it is that the centripetal acceleration is given by the formula v^2/r, and is also subject to Newton's second law: F = ma.

We have the Coulomb force, given in terms of q1 and q2. Well, we're comparing the charges of an electron and proton. You can look up those numbers to get a force. (It will be dependent on r, which is not given)

I'm assuming the gravitational force is negligible in this case. You can look up the mass of an electron and proton to determine the magnitude of the force -- it should be much less than the Coulomb force. (Again, this will be dependent on r)

Now, we have the left side of Newton's law, F. What's the right side, (m*a)?

The question gives you two different periods, 24h, and the hydrogen period. What is a period? It is a certain length of time for something to complete something. What is being completed? A circular orbit around a proton.

So, we have a time given by the problem and a distance, 2(pi)(r). What unit expresses distance and time? Velocity, of course. So, given the information we can construct a formula for the velocity of the electron in terms of the radius of orbit.

In the beginning of the problem I talked about centripetal acceleration, which is related to velocity. Now, you can use this relationship, v^2/r to determine the acceleration, a. The last term m, is simply the mass of an electron.

The last step is just to express Newton's 2nd law, F = ma with all the information we derived. The only term we don't know should be r. After some algebraic manipulation, you can get a general formula for r, the radius of orbit.


Note, this is far from how electrons and protons actually interact. But you can derive a lot of things in the framework of classical mechanics.