Physics 305 Test 2 - Take Home Part

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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.	10 pts.	A cylindrical slab shown in the figure has a polarization that varies linearly with the distance from the central axis of the cylinder (`$s$') $$\vec P (s) = p_0 s \hat s$$ What is the polarization charge inside the slab and what is its bound surface charge?
2.	10 pts.	Find the polarization $\vec P$, the bound charge density $\rho_b$ and the bound surface charge density $\sigma_b$ when a charged cylindrical shell of radius $R$ and carrying a line charge density $\lambda$ ($C/m$) is embedded in a dielectric medium.
3.	20 pts.	Starting with Laplace's Equation find the electric field between two concentric spherical shells of inner radius $a=0.5~m$ and outer radius $b=2.0~m$. The inner sphere is at a potential $V_a = 0~V$ and the outer sphere is at a potential $V_b = 100~V$.

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

4.

20 pts.

A potential difference $V_0$ is applied to a parallel plate capacitor of area $A$ and plate separation $d$ as shown in the figure. The battery is then disconnected and a dielectric slab of thickness $b < d$ and dielectric constant $\epsilon_r$ is placed between the plates as shown. Treat the plates as infinitely large (i.e., $d^2 \ll A$). We have shown that the magnitude of the electric field above an infinite plane of charge is $\vert\vec E\vert = \sigma/\epsilon_0$. Starting with this result and using the definition of the capacitance ($C=Q/V$), what is the capacitance $C_0$ of the plates before the dielectric slab is inserted in terms of $A$, $d$, and any other constants (NOT $V_0$)? What free charge appears on the plates in terms of $A$, $d$, $V_0$, and any other constants? Starting from Gauss's Law, what is the electric field $\vec E_0$ in the gaps between the electric plates and the dielectric slab in terms of $A$, $d$, $V_0$, and any other constants? Starting from Gauss's Law, what is the electric field $\vec E_1$ in the dielectric slab in terms of $A$, $d$, $V_0$, and any other constants? What is the potential difference $V_1$ between the plates after the dielectric slab is inserted in terms of $A$, $d$, $V_0$, and any other constants? Is it bigger or smaller than the starting value? What is the capacitance $C_1$ with the dielectric slab in place in terms of $A$, $d$, $V_0$, and any other constants?