Subfield: Modern Physics

- Proton mass, $m_{p}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Neutron mass, $m_{n}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Electron mass, $m_{e}=9.11 \cdot 10^{-31} \mathrm{~kg}$
- Avogadro's constant, $N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1}$
- Universal gas constant, $R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$
- Boltzmann's constant, $k_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$
- Electron charge magnitude, $e=1.60 \cdot 10^{-19} \mathrm{C}$
- 1 electron volt, $1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J}$
- Speed of light, $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$
- Universal Gravitational constant,

$$
G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2}
$$

- Solar Mass

$$
M_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg}
$$

- Acceleration due to gravity, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$
- 1 unified atomic mass unit,

$$
1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2}
$$

- Planck's constant,

$$
h=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s}
$$

- Permittivity of free space,

$$
\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)
$$

- Coulomb's law constant,

$$
k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2}
$$

- Permeability of free space,

$$
\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}
$$

- Magnetic constant,

$$
\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A}
$$

- 1 atmospheric pressure,

$$
1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa}
$$

- Wien's displacement constant, $b=2.9$. $10^{-3} \mathrm{~m} \cdot \mathrm{K}$
- Stefan-Boltzmann constant,

$$
\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
$$

Question: Consider a gas of mysterious particles called nieons that all travel at the same speed, $v$. They are enclosed in a cubical box, and there are $\rho$ nieons per unit volume. A very small hole of area $A$ is punched in the side of the box. The number of nieons that escape the box per unit time is given by
$$
\alpha v^{\beta} A^{\gamma} \rho^{\delta}
\tag{10}
$$

where $\alpha, \beta, \gamma$, and $\delta$ are all dimensionless constants. Calculate $\alpha+\beta+\gamma+\delta$.

Solution:
The main idea is this: if a nieon is to escape in time $\Delta t$, then it must be traveling towards the hole and be within a hemisphere of radius $v \Delta t$, centered on the hole.
(Note that we will ignore all collisions between nieons. As with many calculations in these kinds of problems, collisions will not affect our answer. Even if they did, we can assume that the hole is so small that we can take $v \Delta t \rightarrow 0$, making the radius of the hemisphere much much smaller than the mean free path.)

First, we will break up this hemisphere into volume elements. We will calculate how many nieons are inside each volume element. Then we will find out how many of these nieons are traveling in the direction of the hole.

We define a spherical coordinate system $(r, \theta, \phi)$. The volume of each differential element is $r^{2} \sin \theta d r d \theta d \phi$, which means that there are $\rho r^{2} \sin \theta d r d \theta d \phi$ nieons inside this volume element.

Within this volume element, how many of these nieons are going towards the hole? We must think about what each nieon "sees." Visualize a sphere of radius $r$ around the volume element. The surface of the sphere will pass through the small hole in the wall. Each nieon is equally likely to go in any direction and is equally likely to end up going through any patch of this spherical surface. Thus, each nieon within this volume element has a probability $A^{\prime} / 4 \pi r^{2}$ of going through the hole, where $A^{\prime}$ is the perceived area of the hole from where the nieon is situated.

$A^{\prime}$ can be calculated as follows. $\hat{\mathbf{j}}$ is the unit vector pointing perpendicular to the hole. Let $\hat{\mathbf{r}}$ be the



unit vector pointing from the hole to the volume element. Then it can be verified that $A^{\prime}=A \cos \psi$, where $\psi$ is the angle between $\hat{\mathbf{j}}$ and $\hat{\mathbf{r}}$.

Expressing $\hat{\mathbf{r}}$ in terms of the other unit vectors, it can be shown that $A^{\prime}=A \sin \theta \sin \phi$. Hence, the fraction of nieons inside the volume element that are going towards the hole is

$$
\frac{A \sin \theta \sin \phi}{4 \pi r^{2}}
$$

Integrating over the entire hemisphere, the number of nieons escaping through the hole in a time $\Delta t$ is

$$
\int_{0}^{v \Delta t} d r \int_{0}^{\pi} d \phi \int_{0}^{\pi} d \theta\left(\frac{A}{4 \pi r^{2}} \sin \theta \sin \phi\right)\left(\rho r^{2} \sin \theta\right)=\frac{1}{4} v A \rho \cdot \Delta t
$$

Final answer: 3.25