Physics 102 Test 3


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

  1. Recall the magnetic lines of force we mapped out in lab for a bar magnet. Does a line of force represent a constant force along its entire length? Explain.

  2. Refer to the diagram below. Explain why, at the two points shown on the circle, the angle between the position vectors $\vec r_1$ and $\vec r_2$ at times \( t_{1} \) and \( t_{2} \) is the same as the angle between the velocity vectors $\vec v_1$ and $\vec v_2$ at times \( t_{1} \) and \( t_{2} \). \includegraphics{circ_motion_fig3.eps}



  3. Consider the result below from our development of the theory behind the mass spectrometer.


    \begin{displaymath} qvB = m\frac{v^2}{r} \end{displaymath}

    For two particles in a mass spectrometer with the same charge and velocity, but different mass, which one will have the larger radius? How will increasing the velocity change the radius? Explain.







  4. It is claimed the solution of the differential equation above describing nuclear decay is the following expression.


    \begin{displaymath} N_{nuc}(t)=N_{0}e^{-\lambda t}\end{displaymath}

    Prove this statement by taking the derivative of \(N _{nuc} (t) \) and showing it satisfies the original differential equation which is the following.

    \begin{displaymath} \frac{dN_{nuc}}{dt}=-\lambda N_{nuc}\end{displaymath}

  5. Recall the discussion of Newton's corpuscular theory of light in the laboratory on interference. Does the data you collected for that lab support Newton's theory or the wave theory? Why?

  6. Suppose it could be shown that the cosmic ray intensity at the Earth's surface was much greater 10,000 years ago. How would this difference affect what we accept as valid carbon-dated values of the age of ancient samples of once-living matter?








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

1. 15 pts. How many half-lives must elapse until 80% of a radioactive sample of atoms has decayed?

2. 17 pts. The uniform magnetic field with $\vert\vec B\vert = 25\times 10^{-3}~T$ in the figure below points in the positive $z$-direction. An electron enters the region of uniform magnetic field with a speed $\vert\vec v\vert = 4\times 10^6~m/s$ and at an angle $\theta = 40^\circ$. What is the radius $r$ and the pitch $p$ of the electron's spiral trajectory? See the figure for the definition of $p$.


\includegraphics[height=3.0in]{eBfield1.eps}

3. 20 pts. Helium atoms emit light at several wavelengths. Light from a helium lamp illuminates a double-slit and is observed on a screen $D=0.50~m$ behind the slits. The emission at wavelength $\lambda_1 = 501.5\times 10^{-9}~m$ creates a first-order bright fringe at a distance $y_1 = 0.219~m$ from the central maximum. What is the wavelength $\lambda_2$ of the bright fringe that is $y_2 = 0.316~m$ from the central maximum?







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



\begin{displaymath} \vec F = m \vec a \quad \vec F_B = q \vec v \times \vec B \q... ...ad \vec F_g = -m \vec g \quad \vert\vec a\vert = {v^2 \over r} \end{displaymath}


\begin{displaymath} \vec E \equiv {\vec F \over q_0} \qquad \Delta V \equiv {\De... ...\int_A^B \vec E \cdot d\vec s \qquad V = k_e {q\over r} \qquad \end{displaymath}


\begin{displaymath} {d N \over d t} = - \lambda t \quad N = N_0 e^{-\lambda t} \quad t_{1/2} = \frac{\rm ln 2}{\lambda} \end{displaymath}


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


\begin{displaymath} E = E_m \sin (kx - \omega t) \quad B = B_m \sin (kx - \omega t) \quad I \propto {Amplitude}^2 \end{displaymath}


\begin{displaymath} I = 4 I_0 \cos^2 \left ( {\pi d \over \lambda} \sin \theta \... ... \delta = d \sin \theta = m \lambda \quad \phi = k\delta \quad \end{displaymath}


\begin{displaymath} \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}} \end{displaymath}


\begin{displaymath} A=\pi r^2 \qquad A = 4\pi r^2 \qquad V = Ah \qquad V = {4\over 3} \pi r^3 \end{displaymath}


\begin{displaymath} \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 \end{displaymath}


\begin{displaymath} {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 \end{displaymath}


\begin{displaymath} \frac{d e^y}{dy} = e^y \qquad \frac{d \log \Omega}{d\Omega} ... ...t_{r_1}^{r_2} \frac{1}{r^2} dr = \frac{1}{r_2} - \frac{1}{r_1} \end{displaymath}


\begin{displaymath} KE_0 + PE_0 = KE_1 + PE_1 \quad KE = {1 \over 2} m v^2 \quad PE = qV \quad \vec p_0 = \vec p_1 \quad p = m\vec v \end{displaymath}