Physics 305 Test 1

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Instructions:

• This part of the exam is closed book, closed note.
• Please show your work. The grade you get depends on how well I can understand your solution even when you write down the correct answer.
• When I grade these problems, I typically give about 60-70% of the points for setting up a problem, and the rest for for doing the algebra, integrations, derivatives, or whatever. Since time may be a factor, I recommend setting up all problems thoughtfully first, then going back to work out the messy stuff.
• If you can't remember something or need a hint to get started on a problem, I may be willing to trade hints for points off your grade. Make me an offer I can't refuse.

Multiple-Choice Questions (5 pts. apiece) Clearly circle the best answer among the different choices.

1. What defines a conservative force?

   A.	$\oint \vec F\cdot\vec a =0$ or $\nabla\cdot\vec F = 0$.	D.	The force must be electromagnetic.
   B.	The force must be frictional.	E.	$\oint \vec F\cdot \vec r =0$ or $\nabla \times \vec F=0$
   C.	The force must be nuclear.

2. Two infinite nonconducting sheets of charge are parallel to each other as shown in the figure. Each sheet has a positive, uniform charge density $\sigma$. Calculate the value of the electric field to the right of the two sheets.

   A.	$0$	D.	$-\frac{\sigma}{\epsilon_0} \hat x$
   B.	$\frac{\sigma}{2\epsilon_0} \hat x$	E.	$\frac{\sigma}{\epsilon_0} \hat x$
   C.	$-\frac{\sigma}{2\epsilon_0} \hat x$

   
   

   

3. A thin rod stretches along the $z$ axis from $z=-d$ to $z=d$ as shown. Let $\lambda$ be the linear charge density or charge per unit length on the rod and the points $P_1 = (0,0,2d)$ and $P_2 = (x,0,0)$. Find the coordinate $x$ such that the potential at $P_1$ is equal to that at $P_2$.

   A.	$0$	D.	$\sqrt{2} d$
   B.	$d$	E.	$2d$
   C.	$\sqrt{3} d$

   
   

   

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

1.	10 pts.	Find the electric field (magnitude and direction) a distance $z$ above the midpoint between two equal charges, $q$, a distance $d$ apart (see figure). Start from the field for a point charge.
2.	10 pts.	A charge $q$ sits at the back corner of a cube of side $a$ as shown in the figure. What is the flux of $\vec E$ through the shaded side? In this problem get your solution to the point where you have a well-defined integral to perform and STOP! Leave your answer in the form of this well-defined integral in Cartesian coordinates.
3.	15 pts.	A hollow, spherical shell carries charge density $$\rho = \frac{k}{r^2}$$ in the region $a \le r \le b$ (see figure). The electric field in the three regions is $$\begin{array}{llll} \vec E & = & \frac{k}{\epsilon_0} \frac{... ...right ) \hat r & a \le r \le b \\ & = & 0 & r < a \end{array}$$ where $k$ is some constant and $\epsilon_0$ is the permittivity of free space. Find the electric potential at the center of the shell using $r$ at infinity as your reference point and starting from the definition of the electric potential in terms of the electric field.

Various and Sundry Equations and Constants

$$\int \frac{1}{x}dx = \log x \quad \int \frac{1}{\sqrt{a^2 + ... ... ) \quad \int \frac{x}{\sqrt{a^2 + x^2}} dx = \sqrt{a^2 + x^2}$$

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

Speed of Light ($c$)	$2.9979\times 10^8~m/s$	proton/neutron mass	$1.67\times 10^{-27}~kg$
$R$	$8.31J/K-mole$	$g$	$9.8~m/s^2$
Gravitation constant	$6.67 \times 10^{-11}~N-m^2/kg^2$	Earth's radius	$6.37\times 10^6~m$
Earth-Moon distance	$3.84\times 10^8~m$	Electron mass	$9.11\times 10^{-31}~kg$