112 年 國立中山大學材料與光電科學學系碩士班《普通物理學》

Problem 1. [Mechanics: 40 points]
In the following, we consider particles only move in x-direction.
(a) [5 points] There are two springs, one with the elastic constant $k_1$, and the other with elastic constant $k_2$. If the two springs are in series, what is the effective elastic constant $k_{eff}$?
(b) [5 points] Consider 2 identical atoms with mass $m$, connected by a chemical bond modeled as a spring with elastic constant $k_0$ (a sigma bond). The center of mass is stationary (not moving). Solve the angular frequency of the vibration motion of the 2 atoms. (hint: it might be easier if using the result of (a))
(c) [5 points] There are two springs, one with the elastic constant $k_1$, and the other with elastic constant $k_2$. If the two springs are in parallel, what is the effective elastic constant $k_{eff}$?
(d) [5 points] Consider 2 identical atoms with mass $m$, connected by 2 chemical bonds (that is, a double bond) modeled as 2 springs with elastic constant $k_0$. The center of mass is stationary (not moving). Solve the angular frequency of the vibration motion of the 2 atoms. (hint: use the result of (c))
(e) [10 points] Consider 2 non-identical atoms with mass $M$ and $m$ (such as a carbon and an oxygen atom), connected by a chemical bond modeled as a spring with elastic constant $k_0$ (a sigma bond). The center of mass is stationary (not moving). Solve the angular frequency of the vibration motion of the 2 atoms.
(f) [5 points] Consider 2 non-identical atoms with mass $M$ and $m$ (such as a carbon and an oxygen atom), connected by 3 chemical bonds modeled as 3 springs with the same elastic constant $k_0$ (a sigma bond). (i) solve the angular frequency of the vibration motion of the 2 atoms, and (ii) compare the obtained frequency with the single bond case above.
(g) [5 points] Carbon dioxide $CO_2$ can be modeled as one carbon atom and two oxygen atoms linked by springs (double bonds). How many are the vibrational modes? Please give a proper argument, not just giving the answer.

Problem 2. [Electromagnetism: 35 points]
The electric field at a distance r from a point charge Q is:
$E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \hat{r}$
where $\hat{r}$ is the unit vector along the vector $\vec{r}$. The electric potential is:
$V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$
Consider the following questions in the vacuum, and use the units in the above formula.
(a) [5 points] If there are two point charges $-q$ and $q$, separated by a distance $d$, as shown in the figure below. The magnitude of the dipole moment $\vec{p}$ of these two point charges is then $p = qd$. Write down the electric potential at $r$ (the origin is at the middle of the two charges), using those quantities given in the figure.
(b) [5 points] Prove $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$.
(c) [5 points] Use the approximation $(1+\delta)^\alpha \approx 1+\alpha\delta$ if $\delta \ll 1$ to show: when $r \gg d$ (measuring the potential far away from the dipole), $V \approx \frac{1}{4\pi\epsilon_0} \frac{qd}{r^2} (1 \pm \frac{1}{2}\frac{d}{r})$.
(d) [10 points] When $r \gg d$ (measuring the potential far away from the dipole), show the electric potential $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$.
(e) [10 points] From the above result, calculate the electric field far away from the dipole using:
$\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial V}{\partial\theta}\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial V}{\partial\phi}\hat{\phi} \right)$

Problem 3. [Thermodynamics: 10 points]
(a) [5 points] A gas is enclosed in a cylinder with a moveable piston, and follows the equation: $p^3V^5 = \text{constant}$.
where $P$ is the pressure, and $V$ is the volume of the gas. Find the work done when the volume is
compressed from $V_A$ to $V_B$, with a known initial pressure $P_A$ (the final pressure $P_B$ is not known).
(b) [5 points] The Maxwell-Boltzmann distribution of the speed v of the particle is:
$f(v) = \left(\frac{m}{2\pi k_B T}\right)^{3/2} 4\pi v^2 e^{-\frac{mv^2}{2k_B T}}$
where $m$ is the mass of the particle, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Calculate
the most probable speed $v_p$, which corresponds to the maximum value of $f(v)$.

Problem 4. [Waves/Optics/Modern physics: 15 points]
If a propagating wave along the x direction has the form $f(x,t) = A e^{i(kx-\omega t+\phi)}$, where $x$ is the
position, $t$ is the time, $A$ is a constant amplitude and $\phi$ is a constant phase.
(a) [5 points] (i) Show that this wave function satisfies the wave equation
$\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$
(ii) express the velocity $v$ using $A$, $k$, $\omega$, or $\phi$.
(b) [5 points] The wave is a periodic function of the time, so $f(x, t_0) = f(x, t_0+T)$ for arbitrary $t_0$
(where $T$ is the period). Use this relation, express the period $T$ with $A$, $k$, $\omega$, and $\phi$.
(c) [5 points] The wave is a periodic function of the position, so $f(x_0, t) = f(x_0+\lambda, t)$ for arbitrary
$x_0$ (where $\lambda$ is the wave length). Use this relation, express the wave length $\lambda$ with $A$, $k$, $\omega$, and $\phi$.