Съдържание

OPhO 2020, Открит кръг — Задача 52. Cramped Up

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от OPhO 2020

2020 Online Physics Olympiad (OPhO): Open Contest v1.5 · 25 май 2020 г.

2020 Online Physics Olympiad (OPhO): Open Contest v1.5

OPhO Lead Organizers and Writers: Adithya Balachandran, Eddie Chen, Ashmit Dutta, Ethan Hu, Viraj Jayam, Rishab Parthasarathy, Kushal Thaman, Max Wang, QiLin Xue

OPhO Website Developer: Ben Oommen

Advisors: Dr. Branislav Kisačanin, Kye Shi, Kevin Zhou

Указания

If you wish to request a clarification, please use this form. To see all clarifications, see this document.

Use $g = 9.81\ \mathrm{m/s^2}$ in this contest. See the constants sheet on the following page for other constants.
This test contains 55 short answer questions. Each problem will have three possible attempts.
The weight of each question depends on our scoring system found here. Put simply, the later questions are worth more, and the overall amount of points from a certain question decreases with the number of attempts and days that you take to solve a problem as well as the number of teams who solve it. This means that your score decreases with the number of tries and days you take to solve a given problem.
Any team member is able to submit an attempt. Choosing to split up the work or doing each problem together is up to you. Note that after you have submitted an attempt, your teammates must refresh their page before they are able to see it.
Answers should contain at least three significant figures, unless otherwise specified. All answers within the 1% range will be accepted.
When submitting a response using scientific notation, please use exponential form. In other words, if your answer to a problem is $A \times 10^B$ , please type $AeB$ into the submission portal.
A standard scientific or graphing handheld calculator may be used. Technology and computer algebra systems like Wolfram Alpha or the one in the TI nSpire will not be needed. Attempts to use these tools will be classified as cheating.
You are allowed to use Wikipedia or books in this exam. Asking for help on online forums or your teachers will be considered cheating and may result in a possible ban from future competitions.
Top scorers from this contest will qualify to compete in the Online Physics Olympiad Invitational Contest, which is an olympiad-style exam. More information will be provided to invitational qualifiers after the end of the Open Contest.
In general, answer in SI units (meter, second, kilogram, watt) unless otherwise specified. Please answer all questions in degrees unless otherwise specified.
If the question asks to give your answer as a percent and your answer comes out to be “ $x\%$ ”, please input the value $x$ into the submission form.
Do not put letters in your answer on the submission portal! If your answer is “ $x$ meters” , input only the value $x$ into the submission portal.
Do not communicate information to anyone else apart from your team-members before the exam ends on May 29, 2020 at 11:59 PM UTC.

List of Constants

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, $R = 8.31\ \mathrm{J/(mol \cdot K)}$
Boltzmann’s constant, $k_B = 1.38 \cdot 10^{-23}\ \mathrm{J/K}$
Electron charge magnitude, $e = 1.60 \cdot 10^{-19}$
1 electron volt, $1\ \mathrm{eV} = 1.60 \cdot 10^{-19}\ \mathrm{J}$
Speed of light, $c = 3.00 \cdot 10^{8}\ \mathrm{m/s}$
Universal Gravitational constant,
$G = 6.67 \cdot 10^{-11}\ \mathrm{(N \cdot m^2)/kg^2}$
Acceleration due to gravity, $g = 9.81\ \mathrm{m/s^2}$
1 unified atomic mass unit,
$1\ \mathrm{u} = 1.66 \cdot 10^{-27}\ \mathrm{kg} = 931\ \mathrm{MeV/c^2}$
Planck’s constant,
$h = 6.63 \cdot 10^{-34}\ \mathrm{J \cdot s} = 4.41 \cdot 10^{-15}\ \mathrm{eV \cdot s}$
Permittivity of free space,
$\epsilon_0 = 8.85 \cdot 10^{-12}\ \mathrm{C^2/(N \cdot m^2)}$
Coulomb’s law constant,
$k = \frac{1}{4\pi\epsilon_0} = 8.99 \cdot 10^{9}\ \mathrm{(N \cdot m^2)/C^2}$
Permeability of free space, $\mu_0 = 4\pi \cdot 10^{-7}\ \mathrm{T \cdot m/A}$
Magnetic constant,
$\frac{\mu_0}{4\pi} = 1 \cdot 10^{-7}\ \mathrm{(T \cdot m)/A}$
1 atmospheric pressure,
$1\ \mathrm{atm} = 1.01 \cdot 10^{5}\ \mathrm{N/m^2} = 1.01 \cdot 10^{5}\ \mathrm{Pa}$
Wien’s displacement constant, $b = 2.9 \cdot 10^{-3}\ \mathrm{m \cdot K}$
Stefan-Boltzmann constant,
$\sigma = 5.67 \cdot 10^{-8}\ \mathrm{W/m^2/K^4}$

Условие

Consider an LC circuit with one inductor and one capacitor. The amplitude of the charge on the plates of the capacitor is $Q = 10\ \mathrm{C}$ and the two plates are initially at a distance $d = 1\ \mathrm{cm}$ away from each other. The plates are then slowly pushed together to a distance $0.5\ \mathrm{cm}$ from each other. Find the resultant amplitude of charge on the parallel plates of the capacitor after this process is completed. Note that the initial current in the circuit is zero and assume that the plates are grounded.

Отговори

Решение

Solution: In slow steady periodic processes (when the time for the change in parameters $\tau$ is much less than the total systems frequency $f$ ), a quantity called the adiabatic invariant $I$ is conserved $^{a}$ . The adiabatic invariant corresponds to the area of a closed contour in phase space (a graph with momentum $p$ and position $x$ as its axes). Note the we can electrostatically map this problem to a mechanics one as the charge corresponds to position, while the momentum would correspond to $LI$ where $I$ is the current and $L$ is the inductance. Thus, in phase space, we have an elliptical contour corresponding to the equation: $\frac{Q^2}{2C} + \frac{(LI)^2}{2L} = C$ where $C$ is a constant in the system. As the area under the curve is conserved, then it can be written that $\pi Q_0 LI_0 = \pi Q_f LI_f$ . It is also easy to conserve energy such that $LI^2 = \frac{Q^2}{C}$ which tells us $I = \frac{Q}{\sqrt{LC}}$ . As $C \propto 1/x$ , we then can write the adiabatic invariant as $xq^4$ which tells us $\boxed{Q_f = \sqrt[4]{2}Q}$ .

We can also solve this regularly by looking at the changes analytically. From Gauss’s law, the electric field between the plates of the capacitors initially can be estimated as

E = \frac{Q}{2\varepsilon_0 A}

where $A$ is the area of the plate. The plates of the capacitor is attracted to the other one with a force of

F = QE = \frac{Q^2}{2\varepsilon_0 A}.

The charges of the plates as a function of time can be approximated as

Q_c = \pm Q\sin(\omega t + \phi).

where $\omega = \frac{1}{\sqrt{LC}}$ . Using this equation, we estimate the average force $\langle F\rangle$ applied on the plate after a period of oscillations to be

\langle F\rangle = \frac{\langle Q^2\rangle}{2\varepsilon_0 A} = \frac{Q^2}{2\varepsilon_0 A}\left\langle\sin^2(\omega t + \phi)\right\rangle = \frac{Q^2}{2\varepsilon_0 A}\cdot\left(\frac{1}{2\pi}\int_0^{2\pi}\sin^2(x)dx\right) = \frac{Q^2}{4\varepsilon_0 A}

this means that after one period, the amount of work done to push the plates closer together is given by

W_F = \langle F\rangle dx = \frac{Q^2}{4\varepsilon_0 A}dx.

In this cycle, the amount of incremental work done by the LC circuit will be given by

dW_{\mathrm{LC}} = \Delta(Fx) = \Delta\left(\frac{Q^2 x}{2\varepsilon_0 A}\right) = \frac{Qx}{\varepsilon_0 A}dQ + \frac{Q^2}{2\varepsilon_0 A}dx.

From conservation of energy, $W_F = W_{LC}$ . Or in other words,

\frac{Q^2}{4\varepsilon_0 A}dx = \frac{Qx}{\varepsilon_0 A}dQ + \frac{Q^2}{2\varepsilon_0 A}dx

simplifying gives us

\frac{Qx}{\varepsilon_0 A}dQ = -\frac{Q^2}{4\varepsilon_0 A}dx

\frac{1}{4}\int\frac{dx}{x} = -\int\frac{dQ}{Q}

\frac{1}{4}\ln x + \ln Q = \mathrm{const.}

We now find our adiabatic invariant to be

xQ^4 = \mathrm{const.}

Substituting values into our equation, we find that

dQ_i^4 = \frac{d}{2}Q_f^4 \implies Q_f = \sqrt[4]{2}Q = \boxed{11.892\ \mathrm{C}}

$^{a}$ This will not be proved in this solution, but the proof can be found in any good mechanics book.

Оригинал в Архива: open.pdf · официални решения: open_sol.pdf