Q26 of 31 Page 7

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


qH= 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 mP


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


mp = 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.


More from this chapter

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24

Fig. 1.11 shows the electric field lines around three point charges A, B and C.

(a) Which charges are positive?


(b) Which charge has the largest magnitude? Why?


(c) In which region or regions of the picture could the electric field be zero? Justify your answer.


(i) near A, (ii) near B, (iii) near C, (iv) nowhere.



 25

Five charges, q each are placed at the corners of a regular pentagon of side ‘a’ (Fig. 1.12).


(a) (i) What will be the electric field at O, the center of the pentagon?


(ii) What will be the electric field at O if the charge from one of the corners (say A) is removed?


(iii) What will be the electric field at O if the charge q at A is replaced by –q?


(b) How would your answer to (a) be affected if pentagon is replaced by n-sided regular polygon with charge q at each of its corners?


 27

Consider a sphere of radius R with charge density distributed as

ρ(r) = kr for r ≤ R


= 0 for r >R .


a) Find the electric field at all points r.


(b) Suppose the total charge on the sphere is 2e where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.


 28

Two fixed, identical conducting plates (α &β) , each of surface area S are charged to –Q and q, respectively, where Q > q > 0. A third identical plate (), free to move is located on the other side of the plate with charge q at a distance d (Fig 1.13). The third plate is released and collides with the plate β. Assume the collision is elastic and the time of collision is sufficient to redistribute charge amongst


(a) Find the electric field acting on the plate before collision.


(b) Find the charges on β and after the collision.


(c) Find the velocity of the plate after the collision and at a distance d from the plate β.