1.13 Assignments 23
be the net change in theyvelocity over the entire collision. Express∆vyin
terms ofvyi,theycomponent ofvi.
(d) Using theycomponent of the equation of motion, obtain a relationship between
dvyanddcosφ.(Hint: it is useful to use conservation of angular momentum to
eliminatedtin favor ofdφ.)Letφiandφfbe the initial and final values ofφ.
By integratingdvy,calculate∆vyover the entire collision. How isφfrelated to
φiand toα(refer to figure)?
(e) How isvyirelated toφiandv∞? How isθrelated toα? Use the expressions
for∆vyobtained in parts (c) and (d) above to obtain the Rutherford scattering
formula
tan
(
θ
2
)
=
q 1 q 2
4 πε 0 μbv^2 ∞
What is the scattering angle for grazing (small angle collisions) andhow does
this small angle scattering relate to the initial center of mass kinetic energy and
to the potential energy at distanceb? For grazing collisions how doesbrelate
to the distance of closest approach? What impact parameter gives 90 degree
scattering?
- One-dimensional Scattering relations:The separation of collision types according to
me/mican also be understood by considering how the combination of conservation of
momentum and of energy together constrain certain properties of collisions. Suppose
that a particle with massm 1 and incident velocityv 1 makes a head-on collision with a
stationary target particle having massm 2 .The conservation equations for momentum
and energy can be written as
m 1 v 1 = m 1 v′ 1 +m 2 v′ 2
1
2
m 1 v^21 =
1
2
m 1 v 1 ′^2 +
1
2
m 2 v 2 ′^2.
where prime refers to the value after the collision. By eliminatingv′ 1 between these
two equations obtainv′ 2 as a function ofv 1. Use this to construct an expression show-
ing the ratiom 2 v 2 ′^2 /m 1 v 12 ,i.e., the fraction of the incident particle energy is trans-
ferred to the target particle per collision. How does this fraction depend onm 1 /m 2
whenm 1 /m 2 is equal to unity, very large, or very small? Ifm 1 /m 2 is very large
or very small how many collisions are required to transfer approximately all of the
incident particle energy to target particles?
- Some basic facts you should know:Memorize the value ofε 0 (or else arrange for the
value to be close at hand). What is the value of Boltzmann’s constant when tempera-
tures are measured in electron volts? What is the density of the air you arebreathing,
measured in particles per cubic meter? What is the density of particles in solid copper,
measured in particles per cubic meter? What is room temperature, expressed in elec-
tron volts? What is the ionization potential (in eV) of a hydrogen atom? What is the
mass of an electron and of an ion (in kilograms)? What is the strength of the Earth’s
magnetic field at your location, expressed in Tesla? What is the strength ofthe mag-
netic field produced by a straight wire carrying 1 ampere as measured byan observer
located 1 meter from the wire and what is the direction of the magnetic field? What
