31. In the Rutherford's experiments on
scattering of a- particles by thin gold foil,
as the impact parameter 'b' increases,
the angle of scattering
O
Increases
Decreases
Is unaffected
May increase or decrease depending on
other factors
Answers
Answer:
In 1911 Ernest Rutherford published a formula which indicated that the number of particles that would be deflected by an angle θ due to scattering from fixed nuclei is inversely proportional to the fourth power of the sine function of one half the angle of deflection; i.e.,
n(θ)Δθ = [κ/sin4(θ/2)] Δθ
where κ is a constant.
It was brilliant analysis in support of brilliant empirical work. Rutherford allowed a beam of alpha particles (helium nuclei) to impinge upon very thin gold foil. The distribution of the deflected alpha particles corresponded to his formula. This result established that the the structure of atoms involved a small dense positively charged nucleus surrounded by the negatively charged electrons. Prior to Rutherford's work the prevalent concept of the structure of atoms was J.J. Thomson's plum pudding model in which the positive and negative charges of atoms were uniformly intermixed.
The following is a derivation of Rutherford's formula with a small degree of generalization.
Consider a body at rest at point O. Another body is coming from the left horizontally initially along the upper red line. It is deflected and travels along a trajectory that asymptotically approaches the line which makes an angle of θ with the horizontal. The problem is to establish the angle of deflection θ. Ernest Rutherford first solved this problem for the electrostatic force. The purpose of this page is present a generalization Rutherford's derivation.
The force between two bodies which is due to a field carried by particles is necessarily of the form f(r)/r² where r is the distance separating the two bodies. For a single field the form of f(r) is Ae−λr, where A are constants for the field. If the field-carrying particles do not decay, as is the case for the photons and gravitons of the electromagnetic and gravitation fields, then λ=0. If the field-carrying particles do decay, as is the case for the π mesons of the nuclear force field, then λ>0.
For a single field the constant A has the form of the product of the field magnitudes of the two bodies, such as masses, charges or number of nucleons, multiplied by a constant characteristic of the field. For example, the force between two bodies of mass m and M is GmM/r², where G is the universal gravitational constant, and thus for this case f(r)=GmM.
For the general case two bodies may be affected by more than one field so f(r) might, for example, be f(r)=JqQ+HnNexp(−λr), where q and Q are the charges of the two bodies and n and N are their nucleon numbers.
For some simple cases the deflection of incoming bodies due to scattering by the target bodies can be analyzed without resort to analyzing the detailed dynamics of the interaction. For simplicity it is assumed that the mass and other characteristics of the incoming bodies, hereafter called particles, are small compared to the target bodies, hereafter called nuclei. This allows the simplification that the target nuclei remain fixed; i.e., do not recoil as a result of the interaction with the incoming particles.
The target nuclei is assumed to be fixed at point O. The incoming particle is following a horizontal trajectory that is located a distance b above the target nuclei. The distance b is commonly termed, the impact parameter for the interaction. Although the particle starts on the horizontal trajectory the interaction with the fields of the nuclei causes it to be deflected and asymptotically approach a line making an angle of θ with the horizontal.
Conservation of Angular Momentum
Let v0 be the velocity of the particle at the initial time and m be its mass. The angular momentum of the particle with respect to the point O is mr0. At a point X on its trajectory the location of the particle can be described by an angle φ and its distance r from the nuclei. At point X the particle has a tangential velocity with respect to O of r(dφ/dt). Thus its angular momentum with respect to O is mr²(dφ/dt). Angular momentum will be conserved therefore
mr²(dφ/dt) = mv0b
which reduces to
dφ/dt = v0b/r²
Change in Linear Momentum
A fundamentatl relation of mechanics is that the change in linear momentum p in any direction over a time interval is equal to the the force F in that direction integrated over the time interval; i.e.,
Δp = p(t1) − p(t0) = ∫t0t1Fdt
Consider the direction shown as OD, which splits the angle (π−θ) in half. The component of the initial linear momentum of mv0 is equal to −mv0cos(β). The angle β works out to be (π/2−θ/2) so the initial component of momentum in the direction OD is, since cos(π/2−θ/2)=sin(θ/2), −mv0sin(θ/2).
By symmetry the terminal velocity will also be v0 and its component of linear momentum in the direction OD will be +mv0sin(θ/2).
Therefore the net change in linear momentum in the direction OD is 2mv0sin(θ/2).
The component of the central force of f(r)/r² in the direction OD is [f(r)/r²]cos(φ).
Thus