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: The following information applies for the next two problems. A circuit has a power source of $\mathcal{E}=5.82 \mathrm{~V}$ connected to three elements in series: an inductor with $L=12.5 \mathrm{mH}$, a capacitor with $C=48.5 \mu \mathrm{F}$, and a diode with threshold voltage $V_{0}=0.65 \mathrm{~V}$. (Of course, the polarity of the diode is aligned with that of the power source.) You close the switch, and after some time, the voltage across the capacitor becomes constant. (Note: An ideal diode with threshold voltage $V_{0}$ is one whose IV characteristic is given by $I=0$ for $V<V_{0}$ and $V=V_{0}$ for $I>0$.)

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How much time (in seconds) has elapsed before the voltage across the capacitor becomes constant?

Solution:
When current is flowing clockwise, the circuit is equivalent to an LC circuit with a power source $\mathcal{E}-V_{0}$. Thus, the voltage $U$ is sinusoidal about its equilibrium voltage $U_{0}=\mathcal{E}-V_{0}$ with frequency $\omega=1 / \sqrt{L C}$.

When the switch is closed, $I=0$ and $U=U_{\min }=0$. Afterwards, $I$ increases to $I_{\max }$ and decreases back to 0 , completing half a period of a sine wave. However, $I$ can not go negative due to the presence of the diode. Instead, a reverse voltage builds up on the diode (so that the voltage across the inductor becomes 0 ), and $I$ stays at 0 . At this point, $U$ becomes constant as well.

The time it took until the system became static was half a period of the LC circuit oscillation, i.e.,

$$
\frac{\pi}{\omega}=\pi \sqrt{L C}=2.446 \times 10^{-3} \mathrm{~s}
$$

Final answer: $2.446 \times 10^{-3}$ s