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In 1959, Lytteton and Bondi suggest that the expansion of the Universe could be explained fi mattercarried a net charge. Supposethat the Universe is made up of hydrogenatoms with a number density N, which is mainted a constant. Let the charge on the proton be , e_(p) = -(1+q) e where e si the electronic charge. (a) Find the critical value of y suchthat expansion may start. (b) Show that the velocity of expansion is propertional to the distance from the center. |
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Answer» Solution :(a) Suppose the universe is a sphere of radius R and itsconstituenthydrogenatoms are distributed uniformly in the sphere. As hydrogenatomcontainsone protonand one electron, therefore, chargeon eachhydrogen atom, `e_(H) = e_(p) + e = -(1+y) e + e = -ye = (ye)` if E electric field intensity at distanceR, on the surfaceof the sphere, then according to Gauss's theroem, `oint vec(E) . vec(ds) = (q)/(in_(0)) , i.e.,E(4pi R^(2)) = (4)/(3) (pi R^(2) N |ye |)/(in_(0))` `E = (1)/(3) (N|ye|R)/(in_(0))`...(i) Now, mass of each hydrogen atom = `m_(p)` = mass of a proton. If `G_(R)` is gravitationalfield at distnaceR on the surface of the sphere, then `-4 pi R^(2) G_(R) = 4 pi G, m_(p) ((4)/(3) pi R^(3) N) or G_(R) = -(4)/(3) pi G m_(p) NR`...(ii) `:.` Gravitational force on this atom is`F_(1) = m_(p) xx G_(R) = - (4pi)/(3) Gm_(p)^(2) N R`...(iii) andcoulomb force on hydrogen atom at R is`F_(2) = (ye) E = (1)/(3) (N y^(2) e^(2) R)/(in_(0))` using (i) The expansion would start when coulomb repulsion `(F_(2))` on hydrogen atom is larger than the gravitationalforce of attraction `(F_(1))`on the hydrogen atom. The critical value of y to start expansion would be when `F_(2) = F_(1) i.e.,(1)/(3) (N y^(2) e^(2) R)/(in_(0)) = (4pi)/(3) G m_(p)^(2) NR` or`y^(2) = (4pi in_(0)) G ((m_(p))/(e))^(2) = (1)/(9xx10^(9)) = (6.67xx10^(-11)) ((1.66xx10^(-27))^(2))/((1.6xx10^(-19))^(2)) = 79.8xx10^(-38)` `y = SQRT(79.8xx10^(-38)) = 8.9xx10^(-19) = 10^(-18)` This is the critical value of y correspondingto whichexpansion of universe would start. (b) Net forceexperienced by the hydrogen atom, `F = F_(2) - F_(1)= (1)/(3) (N y^(2) e^(2) R)/(in_(0)) - (4pi)/(3) Gm_(p)^(2) NR` If acceleration of hydrogen atom is represendedby `d^(2)R// dr^(2)`, then `m_(p) (d^(2)R)/(dr^(2)) = F = (1)/(3) (N y^(2)e^(2))/(in_(0)) R - (4pi)/(3) Gm_(p)^(2) N R = ((1)/(3) (N y^(2) e^(2))/(in_(0)) - (4pi)/(3) G m_(p)^(2) N) R` `:. (d^(2)R)/(dr^(2)) = (1)/(m_(p)) [(1)/(3) (N y^(2) e^(2))/(in_(0)) - (4pi)/(3) G m_(p)^(2) N]R = alpha^(2) R` where `alpha^(2) = (1)/(m_(p)) [(1)/(3) (N y^(2) e^(2))/(in_(0)) - (4pi)/(3) G m_(p)^(2) N]` The genralsolution of EQN. (iv) is`R = A e^(alpha t) + B e^(-alpha t)` As we are lookingfor expansion, `B = 0. :.R =A e^(alpha t)` velocity of expansions, `v = (dR)/(dr) = A e^(alpha t) (alpha) = alpha A e^(alpha t) = alpha R` Hence, `vprop R`, whichwas to be proved. |
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