Physics 102 Test 2

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Questions (8 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. The plot below shows the entropy for two Einstein solids alone ($S_A$ and $S_B$) and in contact ($S_{AB}$) as a function of $q_A$, the number of quanta in Solid $A$. What is $q_A$ for the most probable macrostate? Mark that point on the plot. What mathematical condition can you impose on the total entropy $S_{AB}$ to determine the most probable macrostate? How are the temperatures of solids $A$ and $B$ are related at the most probable microstate?

   
   

   

2. Recall the electroscope we used in lab with the thin metal leaf attached to a post. Suppose you did the following. (1) Rub a rubber rod with wool and bring the rod near the top of the post of the electroscope, but WITHOUT touching the post with the rod. (2) Without moving the rod, touch the top of the post with your hand for a few seconds and then remove your hand. (3) Now remove the rod. What is the charge, if any, on the electroscope? Explain your reasoning.

3. What do the equipotentials and electric field lines look like between two long, charged plates like the ones in the figure? Explain your reasoning.

   
   
   
   

   

4. Consider our calculation of the electric potential for a ring of charge. If we calculate $V$ along the axis of the ring using the known electric field, we confront the following integral.
   
   $$V = - \int_\infty^r \frac{k_e Qx}{ (x^2 + a^2)^{3/2} } dx$$
   
   
   What factors in the integrand can be pulled out in front of the integral? Explain.

5. From Coulomb's Law and the definition of the electric potential, we expect the spatial variation of the potential of a point charge to obey a power law: , where and are constants. What do you predict the value of to be? Why?

6. A uniform electric field is parallel to the $x$ axis. In what direction can a charge be displaced or moved in this field without any external work being done of the charge? Explain.

Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1.	15 pts.	Imagine that the entropy of a certain substance as a function of $N$ and $U$ is given by the formula $S = Nk_b \ln U$. Using the definition of temperature, show that the thermal energy of this substance is related to its temperature by the expression $U=Nk_b T$. What is the molar specific heat of the substance?
2.	17 pts.	What are the strength and direction of the electric field at the position indicated by the dot in terms of $q$, $d$, and any other necessary constants? Give your answer in component form.

3.

20 pts.

Two large plates with charge density $\sigma$ are spaced $d=2.0\times 10^{-3}~m$ apart and form a parallel plate capacitor (essentially just two plates that line up with each other a distance $d$ apart). The electric field between the plates is $\vert\vec E \vert = 5.0 \times 10^5~V/m$. a. What is the expression for the electric field between the plates in terms of $\sigma$, $d$, and any other necessary constants? b. Starting from your result for part a, find an expression for the voltage across the plates. c. What is the voltage across the plates? Calculate a number here. c. What is the charge density on each disk? Calculate a number here.

Some constants and conversion factors.

$\rho$ (water)	$1.0\times 10^3 kg/m^3$	$P_{atm}$	$1.05\times 10^5 ~N/m^2$
$k_B$	$1.38\times 10^{-23}~J/K$	Speed of light ($c$)	$3.0\times 10^{8}~m/s$
$R$	$8.31J/K-mole$	$g$	$9.8~m/s^2$
$0~K$	$\rm -273^\circ~C$	$1 ~ u$	$1.67\times 10^{-27}~kg$
Gravitation constant ($G$)	$6.67 \times 10^{-11}~N-m^2/kg^2$	Earth's radius	$6.37\times 10^6~m$
Coulomb constant ($k_e$)	$8.99\times 10^{9} {N-m^2 \over C^2}$	Electron mass	$9.11\times 10^{-31}~kg$
Elementary charge ($e$)	$1.60\times 10^{-19}~C$	Proton/Neutron mass	$1.67\times 10^{-27}~kg$
Permittivity constant ($\epsilon_0$)	$8.85\times 10^{-12} {kg^2\over N-m^2}$	$\rm 1.0~eV$	$1.6\times 10^{-19}~J$
$\rm ^4 He$ mass	$6.68 \times 10^{-27}~kg$	$\rm ^4 He$ charge	$3.2\times 10^{-19}~C$
$\rm 1 ~MeV$	$10^6 ~ eV$

Physics 102 Equations

$$\vec F = m \vec a = {d\vec p \over dt} \quad \vec a = \frac{... ... ME_0 = ME_1 \quad \vec p = m \vec v \quad \vec p_0 = \vec p_1$$

$$E_{atom} = (n_x + n_y + n_z) \hbar \omega_0 \qquad E = \su... ...ar \omega_0 \qquad \Omega(N,q) = \frac{(q+3N-1)!}{q! (3N-1)!}$$

$$S = k_B\ln \Omega \qquad \frac{1}{T} = \frac{dS}{dE} \qqua... ...0} \qquad C = \frac{1}{n} \frac{dE}{dT} \qquad E = 3Nk_B T$$

$$\vert\vec F_C\vert = k_e {\vert q_1\vert \vert q_2\vert \ove... ...^2} \hat r_i \qquad \vert\vec E\vert = \int {k_e dq \over r^2}$$

$$\vert\vec E\vert = 2 \pi k_e \sigma \qquad \vert\vec E\vert... ...a s \qquad \vec E = -k_e \frac{q d}{(x^2 + d^2/4)^{3/2}}\hat j$$

$$W \equiv \int \vec F \cdot d\vec s \qquad \Delta V \equiv {\... ...\int_A^B \vec E \cdot d\vec s \qquad V = k_e {q\over r} \qquad$$

$$V = k_e \sum_n {q_n \over r_n} \qquad V = k_e \int {dq \over r} \qquad$$

$$x = \frac{a}{2} t^2 + v_0 t + x_0 \qquad v = at + v_0$$

$$\langle x\rangle = \frac{1}{N}\sum_i x_{i} \qquad \sigma = \... ...um_i \left( x_{i}-\left\langle x\right\rangle \right)^2}{N-1}}$$

$$A=\pi r^2 \qquad A = 4\pi r^2 \qquad V = Ah \qquad V = {4\over 3} \pi r^3$$

$$\frac{df}{dx} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Del... ...= \lim_{\Delta x \rightarrow 0} \sum_{x_0}^{x_1} f(x) \Delta x$$

$${d f(u) \over dx} = {df\over du}{du \over dx} \qquad \int_{x_0}^{x^1} \frac{df}{dx} dx = f(x_1) - f(x_0) \qquad$$

$$\frac{d e^y}{dy} = e^y \qquad \frac{dx}{dx} = 1 \qquad \frac... ...t_{r_1}^{r_2} \frac{1}{r^2} dr = \frac{1}{r_2} - \frac{1}{r_1}$$

$$\int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln(x + \sqrt{x^2 \pm a... ...rac{x dx}{(x^2 \pm a^2)^{3/2}} = -\frac{1}{\sqrt{x^2 \pm a^2}}$$

$$\int \frac{dx}{(x^2 \pm a^2)^{3/2}} = \pm\frac{x}{a^2\sqrt{x^2 \pm a^2}}$$