For scattering by an inverse-square field (such as that produced by a charged nucleus in Rutherford's model) the relation between impact parameter b and the scattering angle `theta` is given by, the scattering angle for `b=0` is

Explanation

Given relation: b = `(Ze^(2) cot theta //2)/(4 pi epsilon_(0) ((1)/(2) mv^(2)))`
`0 = (Ze^(2) cot theta //2)/(4 pi epsilon_(0)((1)/(2) mv^(2))) " or " cot (theta)/(2) = 0 cot (theta)/(2) = 0 therefore (theta)/(2) = 90^(@) " or " theta = 180^(@)`
Thus the scattering angle is `180^(@)` when impact parameter is zero.
(b) For given b, `(Ze^(2) cot theta //2)/(4 pi epsilon_(0) ((1)/(2) mv^(2))) ` = constant
As the energy `(mv^(2) //2)` increases, the value of cot `theta`/2 increases. Therefore, the value of scattering angle `theta` decreases as expected.
(c ) `theta = 90^(@) `, Z = 79, e = ` 1.6 xx 10^(-19)` C
now E = `(1)/(2) mv^(2)= 10 ` Me V
`10 xx 10^(6) xx 1.6 xx 10^(-19) J = 1.6 xx 10^(-12) J `
`therefore b = (9 xx 10^(9) xx 79 xx (1.6 xx 10^(-19))^(2) cot 45^(@))/(1.6 xx 10^(-12)) = 1.1 xx 10^(-14)` m
(d) the scattering of `alpha`-particles takes place due to charge on the nucleus. If Z = 0 , `theta = 0^(@)` (from the given formula).
Mass of nucleus does not appear in the expression for b because the recoil of the nucleus is being ignored i.e., the nucleus is assumed to be at rest during its interaction with the `alpha`-particle.
(e) For a given energy `(mv^(2)//2)` of the projectile, the decrease in the value of impact parameter means a decrease in the value of cot`theta`/2 and hence an increase in the value of scattering angle `theta`.