Physics 305 Test 1 - Take Home Part

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

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Instructions:

On this exam, you may use ONLY:	You may NOT use:
$\bullet$ Griffiths (the book, not the dude).	Any other written notes.
$\bullet$ Any other mathematics texts, for instance with integral tables or vector formulas. Web-based math tables are okay too.	Any other physics books.
$\bullet$ Any computer or calculator you like, including Mathematica or the equivalent.	Any other people except me during business hours.
$\bullet$ Your class notes.
$\bullet$ Your own completed homework.
$\bullet$ My homework solutions.

Other Instructions:

• If you have any questions about the rules for this exam, please ask me before taking the exam.
• If you have any questions about the rules after taking the exam, please ask me anyway. If you make an honest mistake about the rules, then at worst we make a small adjustment in your score to remove any unfair advantage you may have had. If you remain silent, it becomes a much more serious honor code violation.

Maximum Time = 120 Minutes

Do it all at once.

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

1.	20 pts.	The electric field in the atmosphere at the Earth's surface is about $200~N/C$ directed downward. At a height $h_1=1400~m$ above the surface of the Earth the electric field is only about $20~N/C$ and still downward. What is the average charge density in the atmosphere below a height $h_1$? Is this charge predominately positive or negative? Assume all the charge is in the region from the Earth's surface to a height $h_1$.
2.	20 pts.	A cylindrically symmetric charge distribution is infinitely long and has a charge density $\rho = C/s$ where $s$ is the distance from the axis of symmetry. What is $\vec E$ as a function of $s$? Let $V=0$ at $s=s_0$. What is $V$ at any other radius?
3.	20 pts.	Imagine that new and precise measurements have revealed an error in Coulomb's law. The true force between two point charges is found to be $$\vec F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{\ensuremath... ...mathcalligra{r}}/\lambda} \hat{\ensuremath{\mathcalligra{r}}~}$$ where $\lambda$ is a new constant of nature. You are charged with rewriting some of the basic laws of electrostatics to accomodate this new force. Assume superposition still holds. What is the electric field of an arbitrary charge distribution? What is the potential of a point charge? Use $\infty$ as your reference point.