Electromagnetic Homework Help
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This document provides an overview of electromagnetic theory focusing on problems related to electrostatic potential, variable separation techniques, and electromagnetic resonances in different geometries like rectangular cavities and waveguides. It discusses boundary conditions, potential calculations, and resonance frequencies, along with specific problems and solutions for rectangular and coaxial geometries. The emphasis is on employing mathematical methods like Fourier series and wave equations to analyze and solve complex electromagnetic scenarios.
Electromagnetic Homework Help
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- 2. ELECTROMAGNETIC THEORY Variable separation In the problems formulated in terms of electrostatic potential one often has to analyze Poisson equation with appropriate boundary conditions on a surface of a conductor or a dielectric There is a popular approach to this problem which starts with constructing a complete set of solutions of the Laplace’s equation in the same geometry and then using it as a basis in the space of functions in order to represent r in question Out of many geometry types amenable to the variable separation technique the three most important ones are the rectangular spherical and cylindrical cases In Problem we review the main ideas of this approach for rectangular geometry using material from electrostatics and then proceed with applying it to EM waves in resonant cavities and transmission lines 1. Variable separation in electrostatics Consider electric field in a rectangular slit 0<x< a, 0 < y < ∞ with the boundaries x = 0,a being ideal conductors a) Suppose that the boundary y = 0 is also an ideal conductor at apotential V while the sides x a are grounded Write down the potential inside the slit as a sum of terms corresponding to standing waves across the slit eduassignmenthelp.com PROBLEMS:
- 3. Using the orthogonality property of the standing waves and their completeness show that any function of x and y can be represented in the form Show that the Laplace’s equation for x y yields ordinary differential equations for the coeffcients . By solving those equations with appropriate boundary conditions obtain the potential (x, y ) b) Now suppose that the boundary y carries a frozen constant charge density while the other two boundaries are ideal grounded conductors as above Solve the problem for by generalizing/modifying the approach of part a). c ) Consider a cube a a a with ideal conducting boundary One face of a cube is held at a potential V while the other five faces are grounded Using Fourier series similar to (1) the potential r within the cube Determine the value at the cube center (numerically or analytically) eduassignmenthelp.com
- 4. 2.Boundary condition for a nearly-perfect conductor Consider a at surface of a ohmic conductor in the presence of electromagnetic field of frequency w « o the conductivity Show that near the surface the components of the fields E and B are related by where n is the normal vector pointing in the conductor This is the boundary condition on the conductor surface for electromagnetic waves written in terms of socalled surface impedance In the perfect conductor limit w/o→ 0, we have Ҫ = 0 Show that in this case the eld components obey the boundary conditions E = 0, B = 0 3.EM resonances of a rectangular cavity Consider a rectangular cavity a b c with boundaries being ideal conductors To nd resonance frequencies determine when the cavity can hold an EM eld with a harmonic time dependence E B eit The eld obeys the wave equation in the domain 0< x < a, 0 < y < b, 0 < z < c, inside the cavity similar for B with the boundary conditions Ek B at the surface a Show that a family of solutions to this problem can be constructed using products of standing waves of the form eduassignmenthelp.com
- 5. and similar for other components of E where a b c for kx(y z ) Find the magnetic eld using Faradays law and verify that the boundary conditions are fullled Find the relation b e t ween dierent components of the electric eld b) From the above determine the spectrum of the resonance frequencies of the system 4.Properties of EM resonances a) Consider a resonance mode in a cavity of an arbitrary shape Show that the time- averaged electric and magnetic field energies are equal. (Hint starting from use the wave equation to express 2 Ec2 in terms of the Laplacian of E and relate r E with B b) For two EM resonance modes with dierent frequencies 1 2 prove the orthogonality RR property (Hint use the same method as in part a). ) 5. Rectangular waveguide Consider electromagnetic waves in a waveguide of a rectangular crossection a b with perfectly conducting walls There are two kinds of waves TE and TM The TE TM modes have electric magnetic eld transverse to k respectively a Show that i The TE modes are given by the solutions to the boundary value problem for the magnetic eld eduassignmenthelp.com
- 6. component parallel to the wavevector with the boundary condition B k n ii The TM modes are given by the solutions of the boundary value problem for the parallel comp o n e n t of electric field with the boundary condition Ek b )Find the TE and TM modes for this problem Give expressions for the cuto frequencies 6.Coaxial transmission line Consider a coaxial cable made of a long cylindrical conducting shell of inner radius a and a wire of radius b on the axis inside The cable is used as a transmission line in the operating TEM mode for which b o t h the electric and magnetic eld are transverse to the wavevector a) Find the elds E B for this mode and show that the dispersion relation for this mode is ck as in vacuum b) Now the cable of length L is made a part of an RC circuit as shown in the gure In this case the cable represents an extended capacitance C Consider how the transmission line is being charged after the switch is closed and the battery voltage V is applied to the cable end Closing the switch creates a steplike E M p u l s e of strength V which propagates from the battery end to the resistor where it is partially reected back then again reected from the battery end and so on The process can b e understood using the results derived for the d wave equation HW Problem with appropriate boundary conditions describing the eld at the ends of the cable Find the potential on the load resistor R as a function of time Show that the time dependence can be described as ringing with frequency cL compare to the case of a eduassignmenthelp.com
- 7. Consider the ringing time dependence separately the cases when the load resistance R is large and small compared to the inverse light s p e e d c At what value of R the ringing is absent This is called an impedance matched load/cable system eduassignmenthelp.com