A hydrogen atom can be considered as having a central point-like proton of positive charge and an electron of negative charge that is distributed about the proton according to the volume charge density . A is a constant, is the Bohr radius, and r is the distance from the centre of the atom. (a) We have to find A using the fact that hydrogen is electrically neutral. (b) We shall then find the electric field produced by the atom at the Bohr radius.

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(a)

The volume charge density of the electron cloud is given by the function

.

Total charge of the electron will be given by the integral

.

We will calculate the integral

.

We make the substitution

We have

And

Therefore, the total charge of the electron cloud will be

.

Or

(b)

We next calculate the total charge contained inside a sphere of radius r centred on the proton. It will be

By applying Gauss’ law,

,

we find that