Physics 305 Test 2

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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. A capacitor is constructed of two square metal plates of area $L^2$ separated by a distance $d$. One half of the space between the plates is filled with a substance of dielectric constant $\kappa_1$. The other half is filled with another dielectric substance with constant $\kappa_2$. Calculate the capacitance of the device assuming the free space capacitance is $C_0$.

   A.	$0.5 C_0 \kappa_1\kappa_2/(\kappa_1 + \kappa_2)$	D.	$2 C_0 \kappa_1\kappa_2/(\kappa_1 + \kappa_2)$
   B.	$(\kappa_1 + \kappa_2)C_0$	E.	$(\kappa_1 + \kappa_2)C_0/2$
   C.	$\kappa_1\kappa_2 C_0/(\kappa_1 + \kappa_2)$

   
   

   

2. Use the fundamental concepts of electromagnetism to determine the electric field of an electric dipole $\vec p$ at a distance $\vec r = r \hat r$.

   A.	$k(3\hat r \cdot \vec p \hat r - \vec p)/r^3$	D.	$k(3\hat r \cdot \vec p \hat r - \vec p)/r^2$
   B.	$k\hat r \cdot \vec p /r^3$	E.	$k(2\hat r \cdot \vec p \hat r - \vec p)/r^3$
   C.	$k\hat r \cdot \vec p /r^2$

   
   

   

3. The electric potential of a grounded conducting sphere of radius $a$ in a uniform electric field is given as $\phi(r,\theta) = -E_0 r[1 - (a/r)^3]\cos\theta$. Find the surface charge distribution on the sphere.

   A.	$\epsilon_0 E_0\sin\theta$	D.	$3\epsilon_0 E_0\cos\theta$
   B.	$\epsilon_0 E_0\cos\theta$	E.	$2\epsilon_0 E_0\sin\theta$
   C.	$2\epsilon_0 E_0\cos\theta$

   
   

   

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

1.	10 pts.	A sphere of radius $R$, centered at the origin, carries charge density $$\rho (r,\theta) = k\frac{R}{r^2} (R-2r)\sin\theta$$ where $k$ is a constant, and $r$, $\theta$, are the usual spherical coordinates. Find the approximate potential in the dipole approximation for points on the $z$ axis, far from the sphere.
2.	10 pts.	A thick spherical shell (inner radius $a$, outer radius $b$) is made of dielectric material with a `frozen-in' polarization $$\vec P(\vec r) = \frac{k}{r} \hat r$$ where $k$ is a constant and $r$ is the distance from the center (see figure). There is no free charge in the problem. Find the electric field in all three regions using the expression $$\oint \vec D \cdot d\vec A = Q_{f_{enc}}$$ to find $\vec D$ and then get $\vec E$ from the following. $$\vec D \equiv \epsilon_0 \vec E +\vec P$$
3.	15 pts.	Find an infinite series for the electric potential in the infinite slot shown below if the boundary at $x=0$ consists of two metal strips: one from $y=0$ to $y=a/2$, is held at a constant potential $V_0$, and the other, from $y=a/2$ to $y=a$, is at potential $-V_0$. In other words, get the general solution for this problem and apply the boundary conditions to obtain an infinite series for the electric potential with a single, unknown coefficient for each term in the series. You do NOT need to determine the unknown coefficient.

Various and Sundry Equations and Constants

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

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

$$\int \frac{1}{x^2 + y^2 + (z - z^\prime)^2}dz^\prime = -\fra... ...x^2+y^2}} \quad \int \cos x \sin^2 x dx = \frac{\sin ^3(x)}{3}$$

$$P_0(x) = 1 \quad P_1(x) = x \quad P_2(x) = \frac{1}{2}\left ... ...1 \right ) \quad P_3(x) = \frac{1}{2}\left ( 5x^3 - 3x\right )$$

$$\frac{\partial^2 X(x)}{\partial x^2} = k^2 X(x) \quad \Rightarrow \quad X(x) = Ae^{kx} + Be^{-kx}$$

$$\frac{\partial^2 Y(y)}{\partial y^2} = -k^2 Y(y) \quad \Rightarrow \quad Y(y) = C\sin(ky) + D\cos(ky)$$

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	$9.11\times 10^{-31}~kg$

Physics Equations

$$\vec E = \frac{\sigma}{\epsilon_0} \hat n \quad \sigma = -\... ...c p \equiv \int \vec r^\prime \rho (\vec r^\prime)d\tau^\prime$$

$$V(\vec r) = \frac{1}{4\pi\epsilon_0} \sum_{n=0}^\infty \frac... ...prime)^n P_n(\cos\theta^\prime)\rho(\vec r^\prime)d\tau^\prime$$

$$\sigma_b = \vec P\cdot \hat n \qquad \rho_b = -\nabla\cdot \vec P$$

$$\vec D = \epsilon_0\vec E + \vec P \qquad \nabla \times \vec... ... \qquad \epsilon_r = 1 + \chi_e = \frac{\epsilon}{\epsilon_0}$$