Physics 132-1 Test 2

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

1. How can an electroscope be used to determine the charge of an object?

2. Consider the field lines of an electric dipole shown in the figure. What do the equipotentials look like? Draw a representative set of equipotentials and state clearly what guided you in your drawing.

   

3. In lab, you were asked to calculate the electric field of the ring with total charge $q$ like the one shown in the figure by starting from the electric field of a point charge and integrating over the charge distribution. In the course of that calculation you encountered the following integral
   
   $$\vec E = \int_{charge} \frac{k_e x}{(x^2 + a^2 )^{3/2}} dq ~ \hat{i}$$
   
   
   where $\hat{i}$ is the unit vector in the $x$ direction, $a$ is the radius of the ring, $dq$ is an infinitesimal piece of charge on the ring, $x$ is the position along the axis in the figure, and $k_e$ is the Coulomb constant. Calculate this integral and explain your steps.

   

   
   
   
   
   
   
   
   

4. Can the resistance of a series combination of electrical resistors ever be less than the resistance of the largest resistor? Explain.

5. Consider the $e/m$ experiment where you measured the mass of the electron. You generated the equation
   
   $$qvB = m \frac{v^2}{r}$$
   
   
   where $q$ is the electronic charge, $v$ is the electron speed, $B$ is the magnitude of the magnetic field, $m$ is the electron mass, and $r$ is the radius of the electron's trajectory. Where does the left-hand side of the equation come from? Where does the right-hand side of the equation come from?

6. Consider the following idea for a novel propulsion for a ship or submarine. In this `magnetohydrodynamic drive' seawater flows between the poles of a magnet as shown below and an electric field drives a current through the seawater. The magnetic force on this current propels the water towards the rear of the ship pushing the ship forward. What should be the direction of the applied electric field in the region between the poles of the magnet? Explain. Movie buffs might recognize this drive from the 1990 movie Hunt for Red October starring Sean Connery.

   
   

   

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

1.

15 pts.

When the two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the third astronaut's orbit to be circular and $h=1.10\times 10^5~ m$ above the surface of the Moon. At this altitude the free-fall acceleration of the Moon's gravity is $g_M = 1.52~m/s^2$. The Moon's radius is $R_M = 1.70\times 10^6~m$. What is the third astronaut's orbital speed?

2.	17 pts.	Three point charges are arranged as shown in the figure. What is the electric field created by charges 1 and 2 at the origin? What is the vector force on charge 3? For charges use $q_1 = 3\times 10^{-9}~C$, $q_2 = -1\times 10^{-9}~C$, and $q_3 = 2\times 10^{-9}~C$. For distances use $r_1 = 0.35~m$ and $r_2 = 0.18~m$.
3.	20 pts.	A beam of electrons with kinetic energy $KE$ emerges from a thin-foil `window' at the end of an accelerator tube. There is a metal plate a distance $d$ from the window and perpendicular to the beam direction. See the figure below. What is the minimum magnetic field $\vec B$ needed to deflect the electron beam and prevent it from hitting the plate? How should $\vec B$ be oriented? Get your answer in terms of $KE$, $d$, and any other necessary 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	$3.2\times 10^{-19}~C$
$\rm 1 ~MeV$	$10^6 ~ eV$

Physics 132-1 Equations

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

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

$$W \equiv \int \vec F \cdot d\vec s \qquad \Delta V \equiv {\... ...\int_A^B \vec E \cdot d\vec s \qquad V = k_e {q\over r} \qquad$$

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

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

=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.

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

$$\vec F_B = q \vec v \times \vec B \qquad \vert\vec F_B\vert... ...vB\sin \theta \vert \qquad \vert\vec F_c\vert = m\frac{v^2}{r}$$

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

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

$${dx^n \over dx} = nx^{n-1} \qquad {d f(u) \over dx} = {df\ov... ... \ \over dx} f(x)\cdot g(x) = f{dg \over dx} + g{df \over dx}$$

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

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

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

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