Chapter 10.4, Problem 79AYU

Rutherford’s Experiment In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The figure shows a diagram from Rutherford’s paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola.

(a) Find an equation of the asymptotes under this scenario.

(b) If the vertex of the path of the alpha particles is 10 cm from the center of the hyperbola, find a model that describes the path of the particle.

To determine

To find:

a. The equation of the asymptotes under given scenario.

Answer to Problem 79AYU

Solution:

a. $y=\pm x$

Explanation of Solution

Given:

A scenario wherein, In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil $0.00004$ cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The given figure below shows a diagram from Rutherford’s paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola.

Formula used:

Equation of the hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

Asymptotes: $y=\pm b/a$.

Calculation:

a. Given that the particle are deflected at an angle of $45°$.

Hence the asymptotes is given by $y=\pm x$.

To determine

To find:

b. A model that describes the path of the particle if the vertex of the path of the particles is 10 cm from the center of the hyperbola.

Answer to Problem 79AYU

Solution:

b. $\frac{x^2}{100}-\frac{y^2}{100}=1$, $x\ge 0$

Explanation of Solution

Given:

A scenario wherein, In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil $0.00004$ cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The given figure below shows a diagram from Rutherford’s paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola.

Formula used:

Equation of the hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

Asymptotes: $y=\pm b/a$.

Calculation:

b. The vertex is 10cm from the center of hyperbola.

Therefore, $a=10$.

The slope of the equation for asymptotes is given by $\pm \frac{b}{a}$.

Therefore,

$\frac{b}{a}=1$; $\frac{b}{10}=1$, $b=10$

Using the origin $\left(0,0\right)$ as the center of the hyperbola, the equation is,

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\frac{x^2}{100}-\frac{y^2}{100}=1$, $x\ge 0$