Physics 132-1 Final Exam

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

1. If one writes the equation for the Balmer series for two different spectral lines - once, say for the $H_\alpha$ wavelength $\lambda_\alpha$, and once for the $H_\beta$ wavelength, $\lambda_\beta$, then one can eliminate $R_H$:
   
   $${1 \over {n_\beta^2}} = {1 \over n_f^2} - \left ( {\lambda_\... ...ght ) \left ( {1 \over n_f^2} - {1 \over n_\alpha^2} \right )$$
   
   
   How can you estimate the value of the three $n$ values in the previous equation?

2. Consider the plot below of the effective potential for a macroscopic particle (blue curve) and its line of constant energy (red curve). Describe the motion of the particle in this potential. What restrictions are there on the energy $E$ of a classical particle?

   

3. Recall the rotational collision of a rotating disk and a cylindrical weight dropped onto the disk. Come up with a formula for the rotational inertia, $I$, of the whole system before and after the collision in terms of $I_r$ the moment of inertia of the rotator, $m_d$ and $r_d$ the mass and radius of the disk, $m_w$ and $r_w$ the mass and radius of the weight dropped on the disk, and $r_{dw}$ the distance from the center of the disk to the center of the weight.

4. In solving the Schroedinger equation what requirement or postulate forces us to choose particular energy states (i.e. what causes energy quantization)?

5. Consider what happens when two Einstein solids come in contact. The number of atoms for solid $A$ is $N_A=1$ it contains two quanta of energy so $q_A=2$. Let $N_B = 2$ (two whole atoms!) and $q_B = 1$. Now bring the your solids $A$ and $B$ `together' into a single system. What is the total multiplicity $\Omega_{AB}$ for the combined system with $q_A=2$ and $q_B = 1$ in its initial macrostate?

6. Consider a container of water with ice added to it and placed on a heater. What is the relationship between the temperature and the added heat after the ice has melted, but before the water begins to boil? What is your evidence?

7. How does a mass spectrometer work?

8. Consider the plot below showing equipotentials for the charges in the figure (filled circles are negative, open circles are positive). Draw a full set of of electric field lines with appropriate directions to accompany the equipotentials. Explain your reasoning.

   

9. Consider the measured intensity pattern from a pair of narrow slits of width $a$ separated by a distance $d$. How would you determine $a$ if $\lambda$ is known?

   

10. Can radiocarbon dating be used to measure the age of very old (about a billion years) material?

11. Consider a solid cube made of a good conductor, i.e. a metal. If charge is added to the cube where will it be located at equilibrium? Explain.

12. Two rooms of equal volume are connected by an open passageway, but are maintained at different temperatures. Does each room have the same number of air molecules? Explain.

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

1.

6 pts.

What is the energy of the hydrogen atom electron whose probability density for $L=0$ is represented by the plot shown below? What minimum energy is needed to remove this electron from the atom?

2.	8 pts.	Imagine that a particle has a wave function $$\begin{eqnarray*} \psi(x) & = \sqrt{\frac{2}{a}} e^{-x/a} \quad & {\rm for} \quad x > 0 \\ & = 0 \qquad \qquad & {\rm for} \quad x < 0 \end{eqnarray*}$$ What is the probability the particle will be found in the range $0 < x < a$?
3.	8 pts.	The liquid-drop model of the atomic nucleus predicts that high-energy oscillations of certain nuclei can split the nucleus into two unequal fragments and a few neutrons. The fission products acquire kinetic energy from their mutual Coulomb repulsion. Calculate the electric potential energy (in electron volts) of two spherical fragments from a uranium nucleus having the following charges and radii: $Z_1 = 32e$ and $R_1 = 5.2\times 10^{-15}~m$; $Z_2 = 60e$ and $R_2 = 6.4\times 10^{-15}~m$. Assume that the charge is uniformly distributed throughout the volume of each spherical nucleus and that just before separating each fragment is at rest and their surfaces are in contact. The electrons surrounding the nucleus can be ignored.
4.	10 pts.	A newly-created material has a multiplicity $$\Omega = \beta M E^2$$ where $N$ is the number of atoms in the solid, $E$ is the total, internal energy in the solid, and $\alpha$ is a constant of proportionality. How does the temperature of the new material depend on the internal energy? What is the molar heat capacity for this solid? Could this material really exist? Why or why not?
5.	10 pts.	The figure below shows a some data for intensity versus diffraction angle for the diffraction of an x-ray beam through a crystal acting like a diffraction grating. The beam consists of two wavelengths and the spacing between the planes of atoms is $d=1.88~nm$. What are the two wavelengths?

6.

10 pts.

In the figure below four charges form the corners of a square and four more charges lie at the midpoints of the sides of the square. The distance between adjacent charges on the perimeter of the square is $d$. What are the magnitude and direction of the electric field at the center of the square in terms of $d$, $q$ and any other constants?

Physics 132-1 Constants

$\rho$ (water)	$1.0\times 10^3 kg/m^3$	$P_{atm}$	$1.05\times 10^5 ~N/m^2$
$k_B$	$1.38\times 10^{-23}~J/K$	proton/neutron mass	$1.67\times 10^{-27}~kg$
$R$	$8.31J/K-mole$	$g$	$9.8~m/s^2$
$0~K$	$\rm -273^\circ~C$	$1 ~ u$	$1.67\times 10^{-27}~kg$
Gravitation constant	$6.67 \times 10^{-11}~N-m^2/kg^2$	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	$3.2\times 10^{-19}~C$
$\rm 1 ~MeV$	$10^6 ~ eV$	Earth-Moon distance	$3.84\times 10^8~m$
Premeability constant ($\mu_0$)	$4\pi\times 10^{-7}~T-m/A$	Speed of Light ($c$)	$2.9979\times 10^8~m/s$

Physics 132-1 Equations

$$\vec F = m \vec a = {d\vec p \over dt} \quad KE = {1 \over 2... ...p_1 \quad x = \frac{a}{2} t^2 + v_0 t + x_0 \quad v = at + v_0$$

$$Q = C\Delta T = cm\Delta T = n C_v \Delta T \quad Q_{f,v} = m L_{f,v}$$

$$\Delta E_{int} = Q+W \quad W = \int \vec F \cdot d\vec s \ri... ...d P = {\vert\vec F\vert \over A} \quad PV = Nk_B T = nRT \quad$$

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

$$v_{rms} = \sqrt{\overline {v^2} } \quad C_{V} = {f\over 2} N... ...2} Nk_BT \quad f \equiv {\rm number\ of\ degrees\ of\ freedom}$$

$$E_{atom} = (n_x + n_y + n_z + \frac{3}{2}) \hbar \omega_0 \q... ...bar \omega_0 \quad \Omega(N,q) = \frac{(q+3N-1)!}{q! (3N-1)!}$$

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

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

$$\vert\vec E\vert = 2 \pi k_e \sigma \quad \vert\vec E\vert ... ...- \int_A^B \vec E \cdot d\vec s \quad V = k_e {q\over r} \quad$$

$$V = k_e \sum_n {q_n \over r_n} \quad V = k_e \int {dq \over ... ... \partial y} \quad E_z = - {\partial V \over \partial z} \quad$$

$$I \equiv {dQ \over dt} \quad V = IR \quad P = IV \quad R_{equiv} = \sum R_i \quad {1 \over R_{equiv}} = \sum {1\over R_i}$$

=100000

=2.5in The algebraic sum of the potential changes across all the elements of a closed loop is zero. The sum of the currents entering a junction is equal to the sum of the currents leaving the junction.

$$I = n e v_d A \quad R = \rho \frac{L}{A}= \frac{m_e}{n e^2 ... ...d \langle E \rangle = \frac{1}{2}mv_{rms}^2 = \frac{3}{2}k_B T$$

$$\vec F_B = q \vec v \times \vec B \quad \vert\vec F_B\vert ... ...N \over d t} = - \lambda t \quad N = N_0 e^{-\lambda t} \quad$$

$$y = A \cos (kx - \omega t) \quad k\lambda = 2 \pi \quad \omega T = 2 \pi \quad \frac{\lambda}{T} = c \quad f = \frac{1}{T}$$

$$E = E_m \sin (kx - \omega t) \quad B = B_m \sin (kx - \omega... ...s \vec B \quad S = \frac{E^2}{\mu_0 c} \quad \frac{E_m}{B_m}=c$$

$$I = \frac{E^2}{2\mu_0 c} = 2\rho_{EM} c \quad I = 4 I_0 \cos... ...a \right )}{\frac{\pi a}{\lambda} \sin \theta}\right ]^2 \quad$$

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

$$\delta = d \sin \theta = m \lambda \quad \delta = a \sin \th... ...a \quad \phi = k\delta \quad \sin \theta_R = {\lambda \over a}$$

$${1 \over \lambda} = R_H \left ( {1 \over n_f^2} - {1 \over n... ...quad I = I_{cm} + mR^2 \quad I_0 \omega_0 = I_1 \omega_1 \quad$$

$$-\frac{\hbar^2}{2 m}\left ( \frac{d^2}{d r^2} \right ) \Psi(... ...rac{L^2}{2 m r^2} \Psi(r) + V \Psi(r) = E \Psi(r) \quad E = hf$$

$${dx^n \over dx} = nx^{n-1} \quad \frac{de^x}{dx} = e^x \quad... ... dx} + g{df \over dx} \quad \frac{d \ln x}{dx} = \frac{1}{x}$$

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

$$\frac{d f(x)}{dx} = \lim_{\Delta x \rightarrow 0} \frac{f(x+... ...uad \int x^n dx = \frac{x^{n+1}}{n+1} \quad \int e^x dx = e^x$$

$$\int \frac{1}{\sqrt{x^2 + a^2}} dx = \ln \left [ x + \sqrt{x... ... ] \quad \int \frac{x}{\sqrt{x^2 + a^2}} dx = \sqrt{x^2 + a^2}$$

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

Moments of Inertia