Subfield: Electromagnetism

- 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: Colliding Conducting Slab A thin conducting square slab with side length $s=5 \mathrm{~cm}$, initial charge $q=0.1 \mu \mathrm{C}$, and mass $m=100 \mathrm{~g}$ is given a kick and sent bouncing between two infinite conducting plates separated by a distance $d=0.5 \mathrm{~cm} \ll s$ and with surface charge density $\pm \sigma= \pm 50 \mu \mathrm{C} / \mathrm{m}^{2}$. After a long time it is observed exactly in the middle of the two plates to be traveling with velocity of magnitude $v=3 \mathrm{~m} / \mathrm{s}$ and direction $\theta=30^{\circ}$ with respect to the horizontal line parallel to the plates. How many collisions occur after it has traveled a distance $L=15 \mathrm{~m}$ horizontally from when it was last observed? Assume that all collisions are elastic, and neglect induced charges. Note that the setup is horizontal so gravity does not need to be accounted for.
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Solution:
After the first collision, the charge approaches a constant magnitude. Let's look at a collision with the plate at charge $q$ hitting the $-\sigma$ plate. Since the slab is thin, to keep the conducting surface an equipotential, the charge on the conduting slab has to become $-\sigma s^{2}$. The initial charge doesn't matter because the plates are infinite. So the charge is $-\sigma s^{2}$ after leaving the negative plate and $+\sigma s^{2}$ after leaving the positive plate.
Now the acceleration always has a constant magnitude, which is directed toward the plate it is traveling to, which we can find using $F=m a$,

$$
\left(\sigma s^{2}\right) \frac{\sigma}{\epsilon_{0}}=m a \Longrightarrow a=\frac{\sigma^{2} s^{2}}{\epsilon_{0} m}=7.06 \mathrm{~m} / \mathrm{s}^{2}
$$



Then here we can use the trick of reflecting over the plates, and now it's a basic kinematics problem. The time to travel $L$ is $t=\frac{L}{v \cos \theta}$, and then the distance it travels vertically is

$$
y=v \sin \theta \frac{L}{v \cos \theta}+\frac{1}{2} a\left(\frac{L}{v \cos \theta}\right)^{2}=126.36 \mathrm{~m}
$$

And since $25272 d+d / 2 \approx 87.129$, the number of collisions is 25273 .

Final answer: 25273