(a) Two protons are placed at some separation in vacuum. Find the ratio of electric and gravitational force acting between them.
(b) Two point charges are placed at separation `3 m` in vacuum. What can be the minimum force between them.
(c ) A charge `Q` is to be divided on two objects. What should be the value of the charges in the objects so that the force between them is maximum ?
(d) Two insulating small spheres are rubbed against each other and placed `1.6 cm` apart. If they attract each other with a force of `0.9 N`, how many electrons were transferred from one sphere to the other during rubbing ?

Explanation

(a) `F_(e ) = (1)/(4 pi in_(0)) (q_(1) q_(2))/(r^(2)) = (1)/(4 pi in_(0)) (e .e)/(r^(2))`
`F_(g) = G (m_(1) m_(2))/(r^(2)) = G (m .m)/(r^(2))`
`(F_(e))/(F_(g)) = ((1)/(4 pi in_(0)) (e^(2))/( r^(2)))/((Gm^(2))/(r^(2))) = (1)/( 4 pi in_(0)) (e^(2))/(m^(2)).(1)/(G)`
`= (( 9 xx 10^(9)) (1.6 xx 10^(-19))^(2))/((1.67 xx 10^(-27))^(2)) xx (1)/((6.67 xx 10^(-11))`
`= 1.2 xx 10^(36) ~= 10^(36`
(b) `F = (1)/( 4 pi in_(0)) (q_(1) q_(2))/(r^(2)) = ( 9 xx 10^(9))/(9) q_(1) q_(2) = 10^(9) q_(1) q_(2)`
`F` is minimum if `q_(1) q_(2)` is minimum.
Minimum charge is equal to `e = 1.6 xx 10^(-19) C`.
`(q_(1))_(min) = (q_(2))_(min) = 1.6 xx 10^(-19) C`
`F_(min) = 10^(9) xx (1.6 xx 10^(-19))^(2) = 2.56 xx 10^(-29) N`
(c) Let `Q` is divided into `q` and `Q - q`
`F = (1)/( 4 pi in_(0)) (q(Q - q))/(r^(2)) = (1)/( 4 pi in_(0)) ((qQ - q^(2)))/(r^(2))`
where `(1)/( 4 pi in_(0) r^(2))` is constant.
For `F` to be maximum , `(dF)/(dq) = 0`
`(dF)/(dq) = ((Q - 2q))/(4 pi in_(0) r^(2)) = 0 rArr q = (Q)/(2)`
Two parts are `Q//2 , Q//2`.
(d) During rubbing , one sphere loses electrons and other gains electrons, so spheres are having charge `+ q` and `-q`.
`F = (1)/( 4 pi in_(0)) (q^(2))/(r^(2))`
`0.9 = ( 9 xx 10^(9) q^(2))/((1.6 xx 10^(-2))^(2)) rArr q = 1.6 xx 10^(-7) C`
`n = (q)/(e ) = (1.6 xx 10^(-7))/(1.6 xx 10^(-19)) = 10^(12)`
Number of electrons transferred` = 10^(12)`.