Physics 132-1 Test 3


I pledge that I have neither given nor received unauthorized assistance during the completion of this work.


Signature height0pt depth1pt width3in


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

  1. The position of two-slit interference maxima can be described by


    \begin{displaymath} y_{m}=\frac{\lambda L}{d}m\end{displaymath}

    where $y_{m}$ is the distance of a bright spot from the central maximum (the distance along the slide in this experiment) and $L$ is the distance from the slits to the phototransistor. The quantity $d$ is the slit separation, \( \lambda \) 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.

  2. Consider this question about oscillating charges. Suppose the motion of the charges can be described as an oscillating dipole so the dipole moment as a function of time looks like the figure below. Assume the dipole is aligned with the $z$-axis. What do you expect the electric field to look like as a function of time at some arbitrary distance $r$ away from the source in the $x-y$ plane? What is the direction of the $\vec E$ field? Make a sketch of your answer on the plot and label your curve.

    \includegraphics[height=2.0in]{induction3fig1.eps}

  3. The decay of atomic nuclei is often characterized by a quantity known as the half-life \( \tau \). The half-life is the period of time for one-half of the original sample to disappear via radioactive decay. This statement can be expressed mathematically as $N_{nuc}(t=\tau )=\frac{N_{0}}{2}$. Starting with this expression show that the decay constant \( \lambda \) and the half-life are related by the following equation.


    \begin{displaymath} \tau =\frac{\ln 2}{\lambda }\end{displaymath}

  4. The intensity pattern due to diffraction of light passing through a single slit is


    \begin{displaymath} I_{diff} = I_m \left (\frac {\sin (\frac {\pi a} {\lambda} \sin \theta)} {\frac {\pi a} {\lambda} \sin \theta} \right )^2 \end{displaymath}

    where $a$ is the size of the single slit, \( \theta \) is the angular position of the light, $I_{m}$ is the maximum intensity at the center, and \( \lambda \) is the wavelength of the light. The shape is shown in the figure below. Generate an expression for the angular width of the central maximum in terms of $a$, \( \lambda \), and any other constants you need.



    \includegraphics[height=1.5in]{diffraction_of_light_fig_3.eps}






  5. If the mass of a radioactive sample is doubled, do the activity of the sample and the decay constant increase, decrease, or stay the same. Explain.

  6. In lab we used a red laser to observe two-slit interference. Would the experiment work if we used white light? Explain.

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

1. 15 pts. A beam of bright laser light of wavelength $\lambda = 450~nm = 4.5\times 10^{-7}~m$ passes through a diffraction grating. Enclosing the space beyond the grating is a large screen forming a half-cylinder centered on the grating with the cylinder's axis parallel to the slits in the grating. Eleven bright spots appear on the screen. What are the maximum and minimum values of the slit separation in the diffraction grating?

2. 17 pts. In an experiment on the transport of nutrients in the root structure of a plant, two radioactive nuclides $X$ and $Y$ are used. Initially 3.0 times more nuclei of type $Y$ are present than of type $X$. Two days later there were 5.20 times more nuclei of type $Y$ than of type $X$. Isotope $X$ has a half-life of 1.8 days (equivalent to $\lambda_X = 0.39~d^{-1}$). What is the decay constant $\lambda_Y$ of isotope $Y$?


3. 20 pts. In a joint French-Soviet experiment in the 1980's to monitor the Moon's surface with a light beam, pulsed radiation from a ruby laser ( $\lambda = 690~nm = 6\times 10^{-7}~m$) was directed to the Moon through a reflecting telescope with a mirror radius of $r_1 = 1.3~m$. A reflecting mirror on the Moon behaved like a circular, plane mirror with a radius $r_2 = 0.1~m$, reflecting the light directly back toward the telescope on the Earth. What fraction of the original light energy was reflected back to the Earth by the mirror on the Moon? Assume that all the light energy is in the central diffraction peak and that it is uniformly spread out in that central diffraction peak. The Earth-Monn distance is $R_m = 3.84\times 10^8~m$.

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



\begin{displaymath} {d N \over d t} = - \lambda t \quad N = N_0 e^{-\lambda t} \quad \end{displaymath}


\begin{displaymath} y = A \cos (kx - \omega t) \quad k\lambda = 2 \pi \quad \omega T = 2 \pi \quad \frac{\lambda}{T} = c \end{displaymath}


\begin{displaymath} 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 \end{displaymath}


\begin{displaymath} 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 \end{displaymath}


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


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


\begin{displaymath} KE_0 + PE_0 = KE_1 + PE_1 \quad KE = {1 \over 2} m v^2 \quad PE = qV \end{displaymath}


\begin{displaymath} \vec F=m \vec a \qquad x = \frac{a}{2} t^2 + v_0 t + x_0 \qquad v = at + v_0 \end{displaymath}


\begin{displaymath} {dx^n \over dx} = nx^{n-1} \qquad \frac{de^x}{dx} = e^x \qqu... ... \ \over dx} f(x)\cdot g(x) = f{dg \over dx} + g{df \over dx} \end{displaymath}


\begin{displaymath} \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 \end{displaymath}


\begin{displaymath} \frac{d f(x)}{dx} = \lim_{\Delta x \rightarrow 0} \frac{f(x+... ...elta x} \quad \frac{df(y)}{dx} = \frac{df(y)}{dy}\frac{dy}{dx} \end{displaymath}


\begin{displaymath} \int \frac{1}{\sqrt{x^2 + a^2}} dx = \ln \left [ x + \sqrt{x... ...] \qquad \int \frac{x}{\sqrt{x^2 + a^2}} dx = \sqrt{x^2 + a^2} \end{displaymath}


\begin{displaymath} \int \frac{x^2}{\sqrt{x^2 + a^2}} dx = \frac{1}{2} x \sqrt... ...rac{1}{2} a^2 \ln \left [ x + \sqrt{x^2 + a^2} \right ] \qquad \end{displaymath}