In 1959 Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density N, which is maintained a constant. Let the charge on the proton be: ep = – (1 + y)e where e is the electronic charge.

(a) Find the critical value of y such that expansion may start.

(b) Show that the velocity of expansion is proportional to the distance from the Centre.

a) As given in question we have to assume that universe constituent is hydrogen and further we are assuming that universe is perfect sphere.

Keeping in the mind that fundamental forces like electrostatic forces and gravitation will be the deciding factors.

As we know that hydrogen contains one e^- and one p⁺

So we can get the charge on one hydrogen atom and that is

q_H= ep + e^- = -(1+y)e + e = -ye

Or magnitude wise |ye|

We know the integral form of gauss theorem which is

Here,

E=electric field

ds=small area

Q=charge enclosed

₌ permittivity of free space or vacuum

Number density = N

∴ Total charge will be

Since the universe is considered to be spherical,

∴

∴

So Electrostatic force on a hydrogen atom at distance R will be E × ye

i.e.

as it will be repulsive in nature

Coming to gravitational field

We know that the mass of an atom of hydrogen is equal to mass of proton let it be m_P

Total mass of hydrogen atom

We know gravitational field

The –ive sign suggests it is attractive in nature

So for expansion of world electrostatic force should be greater than gravitational force

i.e.

For minimum case ,

Where G = 6.674×10^-11 Nm² kg^-2

And

m_p = 1.6726219 × 10^-27 kilograms

Solving this we get

b) So now net force experienced by hydrogen atoms is

As we know,

Let, (since it is a constant term)

Where R is position from center.