Physics 102 Final Exam

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Questions (4 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. One can derive a prediction for the intensity pattern due to diffraction of light passing through a single slit. The result is

   
   

   $$I_{diff} = I_m (\frac {\sin (\frac {\pi a} {\lambda} \sin \theta)} {\frac {\pi a} {\lambda} \sin \theta} )^2$$
   
   

   where $a$ is the size of the single slit, is the angular position of the phototransistor relative to the incident beam, $I_{m}$ is the maximum intensity at the center of the diffraction pattern, and is the wavelength of the light. The figure below shows the diffraction pattern has a central maximum with a series of points where the intensity goes to zero at positive and negative angles. When is the expression for the intensity in the equation above equal to zero?

   

   
   
   
   

2. A known mass of water $m_w$ is placed in the calorimeter cup of mass $m_c$ and specific heat $c_c$ and their temperature $T_1$ recorded. A known mass of warm metal pellets $m_i$ at $T_2$ (with unknown specific heat $c_m$) is added to the water. The final temperature of the mixture after reaching equilibrium is $T_3$. What is the complete heat equation in terms of these quantities and any known specific heats?

   
   
   
   
   
   
   
   

3. Consider a gas in a container. Would it violate Newton's Laws or any other physical law if all the particles in the gas collided in such a way that all of the gas particles ended up in the bottom half of the container leaving the top half empty? Explain.

   
   
   
   
   
   
   
   

4. Recall the laboratory on the kinetic theory of ideal gases where you calculated the specific heat of an ideal gas. We claimed this calculation supported the atomic theory. How?

   
   
   
   
   
   
   
   

5. Is the gravitation constant $g$ (the acceleration of gravity at the Earth's surface) really a constant? Explain.

   
   
   
   
   
   
   
   

6. Find some equipotential surfaces for the charge configuration shown below, which consists of two charged metal plates placed parallel to each other. What is the shape of the equipotential surfaces? Explain your method for finding the equipotentials.

   
   
   

   

7. The two magnets shown in the figure are identical. Identify a point on the figure where the magnetic field is zero. Explain how you picked the point.

   
   
   

   

8. Consider an electron entering a uniform magnetic field created by some Helmholtz coils. What is the direction of the magnetic force $\vec F_B$ on the negatively charged electron if $\vec B = B_0 \hat k$ and $\vec v = v_0 \hat j$?

   
   
   
   
   
   
   
   

9. The results for the age of the Shroud of Turin are shown in the table below for the three labs that performed the measurements. The typical uncertainty in these measurements is a standard deviation of 40 years. Are the results of the three laboratories consistent? Explain.

   Laboratory	R	Age (years)
   Arizona	1.20 x 10	662
   Oxford	1.18 x 10	801
   Zurich	1.19 x 10	731

   
   
   
   
   
   
   
   

10. The position of the interference maxima in a double-slit measurement can be described by

    
    

    $$y_{m}=\frac{\lambda D}{d}m$$
    
    

    where $y_{m}$ is the distance of a bright spot from the central maximum (the distance along the slide in this experiment) and $D$ is the distance from the slits to the phototransistor. The quantity $d$ is the slit separation, is the wavelength of the light, and $m$ is the order of the bright spot. Generate an expression for the distance between adjacent bright spots.

    
    
    
    
    
    
    

11. The hydrogen energy level diagram is shown in the figure. If a hydrogen atom is in the $n=3$ state, it will emit light when it loses energy and drops down to another energy state. What state should it jump to in order to emit light with the longest possible wavelength? Explain.

    

12. An airplane flies due west in the northern hemisphere where the Earth's magnetic field points downward and to the north. On which wing, left or right, are some of the conduction electrons moved to the wingtip by the magnetic force on them? Explain.

    
    
    
    
    
    
    
    

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

1.	6 pts.	What are the wavelengths of the first two members of the Lyman series in the spectrum of hydrogen ($n_f = 1$)? What is the series limit for the Lyman series?
2.	8 pts.	You need to use your cell phone which broadcasts a 900 MHz signal, but you're behind two massive, radio-absorbing buildings that have only a 20 m space between them. What is the angular width, in degrees, of the electromagnetic wave after it emerges from between the buildings?

3.	8 pts.	Uranium has two naturally occurring isotopes. $\rm ^{238}U$ has a natural abundance of 99.3% and $\rm ^{235}U$ has an abundance of 0.7%. It is the rarer $\rm ^{235}U$ that is needed for nuclear reactors. The isotopes are separated by forming uranium hexaflouride $\rm UF_6$ which is a gas and then allowing it to diffuse through a series of porous membranes. The $\bf ^{235}UF_6$ has a slightly higher rms speed than $\rm ^{238}UF_6$ and diffuses slightly faster. Many repetitions of this procedure gradually separates the two isotopes. What is the ratio of the rms speed of $\rm ^{235}UF_6$ to $\rm ^{238}UF_6$? The atomic mass of flourine (the `F' in $\rm UF_6$) is 19 u.
4.	10 pts.	The figure below shows a mass spectrometer used to identify the various molecules in a sample by measuring their charge-to-mass ratio $e/m$. The sample is ionized, the positive ions are accelerated (starting from rest) through a potential difference $\Delta V$, and then they enter a region of uniform magnetic field. The field bends the ions into circular trajectories, but after just half a circle they either strike the wall or pass through a small opening into a detector. As the accelerating voltage is slowly increased, different ions reach the detector and are measured. Typical design values are a magnetic field strength $B = 0.20~ T$ and a spacing between the entrance and exit holes $d=0.08~m$. What accelerating potential difference is required to detect $\rm CO_2^+$ where the mass of carbon is 12.0000 u and the mass of oxygen is 15.9949 u?
5.	10 pts.	A newly-created material has a multiplicity $$\Omega = \beta N E e^{\alpha N E}$$ where $N$ is the number of atoms in the solid, $E$ is the total, internal energy in the solid, and $\alpha$ and $\beta$ are constants of proportionality. How does the temperature of the new material depend on the internal energy?

6.

10 pts.

Two $m=0.01~kg$ spheres on a thread of length $d=1.0~m$ repel each other after being charged to $q=10^{-7}~C$. What is the angle $\theta$ in the figure? You can assume that $\theta$ is a small angle so $\sin \theta \approx \theta$ and $\cos \theta \approx 1$.

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 ($k_e$)	$8.99\times 10^{9} {N-m^2 \over C^2}$	Earth's mass	$5.97\times 10^{24}~kg$
Elementary charge ($e$)	$1.60\times 10^{-19}~C$	Proton/Neutron mass	$1.67\times 10^{-27}~kg$
Permittivity constant ($\epsilon_0$)	$8.85\times 10^{-12} {kg^2\over N-m^2}$	Electron mass	$9.11\times 10^{-31}~kg$
$\rm ^4 He$ mass	$6.68 \times 10^{-27}~kg$	$\rm ^4 He$ charge	$3.2\times 10^{-19}~C$
$\rm 1 ~MeV$	$10^6 ~ eV$	$\rm 1.0~eV$	$1.6\times 10^{-19}~J$
Planck's constant ($h$)	$6.63\times 10^{-34}~J-s$	Planck's constant ($h$)	$4.14\times 10^{-15}~Ve-s$
Planck's constant ($\hbar$)	$1.05\times 10^{-34}~J-s$	Planck's constant ($h$)	$6.58\times 10^{-16}~Ve-s$
Boltzmann's Constant ($l_B$)	$1.3807\times 10^{-23}~ J/K$	Rydberg constant ($R_H$)	$1.0974\times 10^7 ~ m^{-1}$

Physics 102 Final

$$\vec F = m \vec a = {d\vec p \over dt} \quad KE = {1 \over 2... ...ec p = m \vec v \quad \vec p_0 = \vec p_1 \quad F_g = mg \quad$$

$$y(t) = -\frac{g}{2}t^2 + v_0t \quad v_y(t) = -gt + v_0 \quad... ...elta T = cm\Delta T = n C_v \Delta T \quad Q_{f,v} = m L_{f,v}$$

$$\Delta E_{int} = Q+W \quad W = {\rm force} \times {\rm dista... ... \langle \vec F \rangle = {\Delta \vec p \over \Delta t} \quad$$

$$P = {\vert\vec F\vert \over A} \quad PV = Nk_B T = nRT \quad N = n N_A$$

$$\vec I = \int \vec F dt = \langle \vec F \rangle \Delta t ... ...\quad E_{int} = N ~ \langle E_{kin} \rangle = {3\over 2} Nk_BT$$

$$v_{rms} = \sqrt{\overline {v^2} } \quad C_{V} = {f\over 2} N... ...B \quad E_f = { k_BT \over 2} \quad E_{int} = {f\over 2} Nk_BT$$

f $\equiv$ number of degrees of freedom

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

$$S = k_B\ln \Omega \quad \frac{1}{T} = \frac{dS}{dE} \quad ... ...a_0} \quad C = \frac{1}{n} \frac{dE}{dT} \quad E = 3Nk_B T$$

$$\vert\vec F_G\vert = G {m_1 m_2 \over r^2} \quad \vert\vec F... ...uad \vert \vec F_B \vert = \vert q v B \sin \theta \vert \quad$$

$$\vert\vec E\vert = 2 \pi k_e \sigma \quad \vert\vec E\vert ... ... = N_0 e^{-\lambda t} \quad t_{1/2} = \frac{\rm ln 2}{\lambda}$$

$$W \equiv \int \vec F \cdot d\vec s \quad \Delta V \equiv {\D... ...e \sum_n {q_n \over r_n} \quad V = k_e \int {dq \over r} \quad$$

$$y = A \sin (kx - \omega t) \quad k\lambda = \omega T =2 \pi ... ...ad B = B_m \sin (kx - \omega t) \quad I \propto {Amplitude}^2$$

$$I = 4 I_0 \cos^2 \left ( {\pi d \over \lambda} \sin \theta \... ...\sin \theta)} {\frac {\pi a} {\lambda} \sin \theta} \right )^2$$

$$E = \frac{1}{2}mv_r^2 + \frac{L^2}{2mr^2} - k_e\frac{e^2}{r}... ...mbda} = R_H \left ( {1 \over n_f^2} - {1 \over n_i^2} \right )$$

$$\langle x\rangle = \frac{1}{N}\sum_i x_{i} \quad \sigma = \s... ...2 \quad A = 4\pi r^2 \quad V = Ah \quad V = {4\over 3} \pi r^3$$

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

$${d f(u) \over dx} = {df\over du}{du \over dx} \quad \int_{x_... ...t_{r_1}^{r_2} \frac{1}{r^2} dr = \frac{1}{r_2} - \frac{1}{r_1}$$