物理代写|PHYS415 Electromagnetism

Download the fortran program “relax.f” and the gnuplot files “V.gnu” and “V2.gnu” from the course web page on a computer with fortran and gnuplot, such as the network in Wong. To compile it, type “f77 relax.f” and to run it type “./a.out”. You are asked to modify this program so that it solves the problem of a charged square conductor inside of a larger uncharged square conducting cavity. You can use the “pico” text editor to edit it. You will find it convenient to change the array bounds so that the exterior boundary is bounded by the corners $( \pm N 1, \pm N 1)$. Define another variable which specifies the corners of the inner boundary. To implement the extra boundary conditions, modify the logical condition in line 34 appropriately, using logical operators like “.not.” and “.or.”. (A useful fortran reference on the web is http://www.obliquity.com/computer/fortran/ – see the “operations” section.) Lines in fortran must start after column 6 and not go past column 72; to make a longer line, denote a continuation by placing any character in column 6 of the next line.

Find the potential when the inner conductor is 5 times smaller on a side than the outer one. You can experiment with different initial conditions for the potential and different sizes for the lattice. How many iterations does it take to converge? To print your final result, you can uncomment the first two lines of “V.gnu” (by removing the initial “#” sign) and run “gnuplot V.gnu”. This will output the graph to a postscript file “V.ps” which can be printed. You should also produce a postscript file showing your modified program using the command “a2ps -o ps_file_name text_file_name”. If you would prefer to convert the fortran program to $\mathrm{C}$ or some other programming language, you may do so.

(a) An infinite slab of dielectric (with constant $\kappa$ ) sits on the $x-y$ plane, in the region $0 \leq z \leq a$. There is an external field $\vec{E}=E_0(\cos \theta, 0, \sin \theta)$ in the absence of the dielectric. What is the total $\vec{E}$ field when the dielectric is present? Sketch the field lines. (b)Is there any limit in which the dielectric behaves like a conductor? How would the solution look if the dielectric was a conductor instead? Is the problem well posed in this case? (c) Suppose a free charge density $\sigma$ is added, which is the same on both surfaces of the dielectric. How does the solution change?

Complete the problem started in lecture: find the potential for an infinite dielectric cylinder of radius $R$ with external line charge density $\lambda$ (parallel to cylinder) at radius $a>R$.

(a) Using known properties of air at room temperature, including the dielectric constant, find the effective value of the polarizability $\alpha$; express it as $\alpha /\left(4 \pi \epsilon_0\right)$ in units of $\mathrm{m}^3$. How does it compare to the polarizability of the $\mathrm{H}$ atom?
(b) Assuming the Clausius-Mossotti formula is correct, at what pressure (in atmospheres) would nonlinear effects start to become important for predicting the susceptibility of air at room temperature relative to its S.T.P. (take presure $=1$ atmosphere and $T=20^{\circ} \mathrm{C}$ ) value? By what factor is the susceptibility already increased, ignoring nonlinearities, at this pressure?

(a) Using the known dipole moment for $\mathrm{H}_2 \mathrm{O}$ molecules, and ignoring the induced polarization, calculate the polarizability of water at room temperature.
(b) Compute the number density of water molecules in liquid water at S.T.P., and the prediction of the Clausius-Mossotti model for the dielectric constant of water. Could our neglect of the induced polarization help to explain why the prediction is not very good?
(c) Turning the calculation around, given the known dielectric constant of water, how large a dipole moment for $\mathrm{H}_2 \mathrm{O}$ does the Clausius-Mossotti model predict?