(a) By equating the number of beam particles per unit time crossing an annulus…

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch Scattering in a central field

Problem

(a) By equating the number of beam particles per unit time crossing an annulus between impact parameters

b

and

b+\mathrm{d}b

with the number per unit time entering the corresponding cone of solid angle between

\chi

and

\chi+\mathrm{d}\chi

, derive the general relation

\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{b}{\sin\chi}\left|\frac{\mathrm{d}b}{\mathrm{d}\chi}\right|.

(b) For an attractive Coulomb potential

V(r)=-k/r

(

k>0

) between two opposite point charges of relative kinetic energy

E

, the deflection has magnitude

\tan(\chi/2)=k/(2Eb)

, the same functional form as for a repulsive Coulomb potential (only

|k|

enters). Using part (a), derive

\mathrm{d}\sigma/\mathrm{d}\Omega

for the attractive case and show it is identical to the Rutherford result for the repulsive Coulomb potential of the same

|k|

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