For scattering by an 'inverse square' field (such as that produced by a charged culeus in Rutherfor's model ), the relation between impact parmeter b and the scattering angle thetais given by : b = (Ze^(2) cot theta//2)/(4 pi epsilon_(0)((1)/(2) mv^(2))) (a)What is the scattering angle for b = 0 ? (b) For a given impact parameter b, does the angle of deflection increase or decrease with increase in energy ? (c) What is the impact parameter at which the scattering angle is 90^(@) for Z = 79 and initial energy equal to 10 MeV? energy? What is the impact parameter at which the scattering angle is 90^(@) for Z = 79 and initial energy equal to 10 MeV? (d) Why is it that the mass of the nucleus does not enter the formula above but its charge does ? (e) For a given energy of the projectile does the scattering anlgle increase or decrease with decrease in impact parameter ?

For scattering by an 'inverse square' field (such as that produced by

1.	For scattering by an 'inverse square' field (such as that produced by a charged culeus in Rutherfor's model ), the relation between impact parmeter b and the scattering angle thetais given by : b = (Ze^(2) cot theta//2)/(4 pi epsilon_(0)((1)/(2) mv^(2))) (a)What is the scattering angle for b = 0 ? (b) For a given impact parameter b, does the angle of deflection increase or decrease with increase in energy ? (c) What is the impact parameter at which the scattering angle is 90^(@) for Z = 79 and initial energy equal to 10 MeV? energy? What is the impact parameter at which the scattering angle is 90^(@) for Z = 79 and initial energy equal to 10 MeV? (d) Why is it that the mass of the nucleus does not enter the formula above but its charge does ? (e) For a given energy of the projectile does the scattering anlgle increase or decrease with decrease in impact parameter ?
Answer» Solution :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 ` `thereforeb = (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`.

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