## Solving for the distribution of charge where time-averaged potential is given

by **Sim P. Bantayan**, MSPhysics I, MSU-IIT

**Problem 1.5**

The time-averaged potential of a neutral hydrogen atom is given by

where q is the magnitude of the electronic charge, and being the Bohr radius. Find the distribution of charge( both continuous and discrete) that will give this potential and interpret your result physically.

**Solution:**

We can solve the distribution of charge by solving first the charge density using the Poisson’s equation

.

We are given a hint to find the discrete distribution of charge, meaning likely a delta function will be in our answer.

Now, we solve Poisson’s equation for ,

where .

But . So,

Using the product rule on the first term, we obtain

Then we distribute the term and we get

And then using product rule, we have

After the terms cancel, we are only left with

Using our delta function equation for the first term,

,

thus

Physically, this is the point charge of the proton nucleus represented by the

delta function at the center of the atom, surrounded by the negative electron

cloud.