Physics 132-2 Test 2


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Questions (9 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

  1. Is the gravitational constant $g = 9.8~m/s^2$ really a constant? Explain.

  2. If the electric potential is zero, then must the electric field be zero as well? Explain.

  3. 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.

    \begin{displaymath} V = - \int_\infty^r \frac{kQx}{ (x^2 + a^2)^{3/2} } dx \end{displaymath}

    What factors in the integrand can be pulled out in front of the integral? Explain.

  4. Consider an oscilloscope like we used to study the magnetic force on moving charges. With the proper settings you saw a single bright dot in the center of the screen of the scope. What, if anything, will happen to the spot on the screen if the south pole of a magnet is brought near the left side of the oscilloscope? Explain your reasoning.

  5. The figure below shows three arrangements of two protons. Rank the arrangements according to the net electric potential produced at point $P$ by the protons, greatest first.



    \includegraphics{f1.eps}

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

1. 15 pts. An object having a net charge $Q=-3.0\times 10^{-5}~C$ is placed in a uniform electric field of $\vert\vec E \vert = 1000~N/C$ directed vertically. What is the mass $m$ of the object if it `floats' in the field? What is the direction of the field?

2. 20 pts. An accelerator produces a beam of $\rm ^4He$ nuclei consisting of two protons and two neutrons so the charge on each nucleus is $q = 2e =3.2\times 10^{-19}~C$ and the mass is $m=6.68\times 10^{-27}~kg$. The beam energy is $E_b = 4.0~MeV$ and its velocity is $v_b = 1.38\times10^7~m/s$. The beam current is $I = 10^{-5}~A$.

a.
How far apart are the $\rm ^4He$ nuclei on average?

b.
The $\rm ^4He$ nuclei are all positively charged so they repel one another. Is this repulsion a significant factor in beam stability? Explain.

3. 20 pts. You are a summer research student at a medical-school lab that uses proton beams to treat cancer patients. The protons exit the machine with a speed $v_0 = 4.0\times 10^5~m/s$ and you've been asked to design a device to stop the protons safely. You can simply have the protons strike a metal plate to stop them, but protons traveling faster than $v_l = 2.0\times 10^5~m/s$ emit dangerous x-rays when they hit. You decide the best thing to do is to slow down the protons to below $v_l$ and then let them hit the metal plate. You take two, flat, parallel, metal plates a distance $d=0.02~m$ apart and drill a small hole through the center of one plate to let the beam through. The second plate will stop the protons.

a.
What is the minimum electric potential across the plates?

b.
What are the minimum charge densities you need to place on each plate?





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 ($\epsilon_0$) $8.85\times 10^{-12} {kg^2\over N-m^2}$ $\rm 1.0~eV$ $1.6\times 10^{-19}~J$
$\rm ^4He$ mass $6.68 \times 10^{-27}~kg$ $\rm ^4He$ charge $ 3.2\times 10^{-19}~C$
$\rm 1 ~MeV$ $10^6 ~ eV$

Physics 132-2 Equations



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


\begin{displaymath} \vert\vec E\vert = 2 \pi k_e \sigma \qquad \vert\vec E\vert = k_e {qz \over (z^2 + R^2)^{3/2}} \end{displaymath}


\begin{displaymath} 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 \end{displaymath}


\begin{displaymath} V = k_e \sum_n {q_n \over r_n} \qquad V = k_e \int {dq \over... ...partial y} \qquad E_z = - {\partial V \over \partial z} \qquad \end{displaymath}


\begin{displaymath} V = \frac{k_e Q}{\sqrt{x^2 + a^2}} \qquad I \equiv {dQ \over... ...uiv} = \sum R_i \qquad {1 \over R_{equiv}} = \sum {1\over R_i} \end{displaymath}


=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.



\begin{displaymath}\vec F_B = q \vec v \times \vec B \qquad \vert\vec F_B\vert = \vert qvB\sin \theta \vert \end{displaymath}


\begin{displaymath} KE = {1 \over 2}mv^2 \qquad \vec F=m \vec a \qquad F_g = mg \qquad ME = KE + PE \end{displaymath}


\begin{displaymath} 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 {d f(u) \over dx} = {df\ov... ... \ \over dx} f(x)\cdot g(x) = f{dg \over dx} + g{df \over dx} \end{displaymath}


\begin{displaymath} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \qquad f(x) = f(a) + ... ...rime (a) (x-a) + \frac{f^{\prime\prime}(a)}{2}(x-a)^2 + \cdots \end{displaymath}


\begin{displaymath} (a + b)^n = a^n + \frac{n}{1!}a^{n-1}b + \frac{n(n-1)}{2!}a^{n-2}b^2 + \cdots \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}