Questions and Answers of Electrodynamics

A 60-? lossless line terminated by load ZL has a voltage wave as shown in figure. Find s, F, and ZL.
The following slotted-line measurements were taken on a 50-Ω system. With load: s = 3.2, adjacent Vmin, occurs at 12 cm and 32 cm (high numbers on the load side); with short circuit: Vmin
A 50-Ω air slotted line is applied in measuring a load impedance. Adjacent minima are found at 14 cm and 22.5 cm from the load when the unknown load is connected and Vmax = 0.95 V and Vmin =
Show that for a dc voltage Vg turned on at t = 0 (see Figure), the asymptotic values (t
A 60-Ω lossless line is connected to a 40-Ω pulse generator. The line is 6 m long and is terminated by a load of 100 0. If a rectangular pulse of width 5μ and magnitude 20 V is sent
The switch in figure is closed at t = 0. Sketch the voltage and current at the right side of the switch for 0
For the system shown in figure, sketch V(?, t) and I(?, t) for 0
Refer to figure, where Zg = 25?, Zo = 50?, ZL = 150?, ? = 150 m, u = c. If at f = 0, the pulse shown in figure is incident on the line(a) Draw the voltage and current bounce diagrams.(b) Determine
A microstrip line is 1 cm thick and 1.5 cm wide. The conducting strip is made of brass (σc = 1.1 X 107 S/m) while the substrate is a dielectric material with εr = 2.2 and tan θ =0.002.
A 50-Ω microstrip line has a phase shift of 45° at 8 GHz. If the substrate thickness is h = 8 mm with εr = 4.6, find: (a) The width of the conducting strip, (b) The length of
An alumina substrate (ε = 9.6εo) of thickness 2 mm is used for the construction of a microstrip circuit. If the circuit designer has the choice of making the line width to be within 0.4 to
Design a 75-Ω microstrip line on a 1.2-mm thick-duroid (εr = 2.3) substrate. Find the width of the conducting strip and the phase velocity.
Show that a rectangular waveguide does not support TM10 and TM01 modes.
A 2-cm by 3-cm waveguide is filled with a dielectric material with εr = 4. If the waveguide operates at 20 GHz with TMU mode, find:(a) Cutoff frequency,(b) The phase constant,(c) The phase velocity.
A 1-cm X 2-cm waveguide is filled with deionized water with εr = 81. If the operating frequency is 4.5 GHz, determine:(a) All possible propagating modes and their cutoff frequencies,(b) The
Design a rectangular waveguide with an aspect ratio of 3 to 1 for use in the k band (18-26.5 GHz). Assume that the guide is air filled.
A tunnel is modeled as an air-filled metallic rectangular waveguide with dimensions a = 8 m and b = 16 m. Determine whether the tunnel will pass:(a) A 1.5-MHz AM broadcast signal,(b) A 120-MHz FM
In an air-filled rectangular waveguide, the cutoff frequency of a TE10 mode is 5 GHz, whereas that of TE01 mode is 12 GHz. Calculate(a) The dimensions of the guide(b) The cutoff frequencies of the
An air-filled hollow rectangular waveguide is 150 m long and is capped at the end with a metal plate. If a short pulse of frequency 7.2 GHz is introduced into the input end of the guide, how long
Calculate the dimensions of an air-filled rectangular waveguide for which the cutoff frequencies for TM11 and TE03 modes are both equal to 12 GHz. At 8 GHz, determine whether the dominant mode will
An air-filled rectangular waveguide has cross-sectional dimensions a = 6 cm and b = 3 cm. Given that calculate the intrinsic impedance of this mode and the average power flow in theguide.
In an air-filled rectangular waveguide, a TE mode operating at 6 GHz has Ey = 5 sin (2πx/a) cos(πy/b) sin (wt – 12z) V/m Determine: (a) The mode of operation, (b) The cutoff
In an air-filled rectangular waveguide with a = 2.286 cm and b = 1.016 cm, the y-component of the TE mode is given by Ey = sin (2πx/a) cos (3πy/b) sin (10π x 1010t – βz)
For the TM11 mode, derive a formula for the average power transmitted down the guide.
(a) Show that for a rectangular waveguide. (b) For an air-filled waveguide with a = 2b = 2.5 cm operating at 20 GHz, calculate up and ? for TE11 and TE21 modes.
A 1-cm X 3-cm rectangular air-filled waveguide operates in the TE12 mode at a frequency that is 20% higher than the cutoff frequency. Determine:(a) The operating frequency,(b) The phase and group
A microwave transmitter is connected by an air-filled waveguide of cross section 2.5 cm X 1 cm to an antenna. For transmission at 11 GHz, find the ratio of (a) The phase velocity to the medium
A rectangular waveguide is filled with polyethylene (ε = 2.25εo) and operates at 24 GHz. If the cutoff frequency of a certain TE mode is 16 GHz, find the group velocity and intrinsic
A rectangular waveguide with cross sections shown in figure has dielectric discontinuity. Calculate the standing wave ratio if the guide operates at 8 GHz in the dominantmode.
A 1-cm × 2-cm waveguide is made of copper (σc = 5.8 × 107 S/m) and filled with a dielectric material for which ε = 2.6εo, μ = μo, σd = 10–4 S/m. If the guide operates at
A 4-cm-square waveguide is filled with a dielectric with complex permittivity εc = 16εo (l – j10–4) and is excited with the TM21 mode. If the waveguide operates at 10% above the cutoff
If the walls of the square waveguide in the previous problem are made of brass (ac = 1.5 × 107 S/m), find ac and the distance over which the wave is attenuated by 30%.
A rectangular waveguide with a = 2b = 4.8 cm is filled with teflon with εr =2.11 and loss tangent of 3 X 10–4. Assume that the walls of the waveguide are coated with gold (σC = 4.1 X
A rectangular brass (σc = 1.37 X 107 S/m) waveguide with dimensions α = 2.25 cm and b = 1.5 cm operates in the dominant mode at frequency 5 GHz. If the waveguide is filled with teflon (μr = 1, εr
For a square waveguide, show that attenuation αc. is minimum for TE10 mode when f = 2.962 fc.
The attenuation constant of a TM mode is given byAt what frequency will ? be maximum?
Show that for TE mode to z in a rectangular cavity, Find Hxs.
For a rectangular cavity, show thatfor TM mode to z. DetermineEys.
In a rectangular resonant cavity, which mode is dominant when (a) α < b < c (b) α > b > c (c) α = c > b
For an air-filled rectangular cavity with dimensions α = 3 cm, b = 2 cm, c = 4 cm, determine the resonant frequencies for the following modes: TE011, TE101, TM110, and
A rectangular cavity resonator has dimensions α = 3 cm, b = 6 cm, and c = 9 cm. If it is filled with polyethylene (ε = 2.5εo), find the resonant frequencies of the first five
An air-filled cubical cavity operates at a resonant frequency of 2 GHz when excited at the TE101 mode. Determine the dimensions of the cavity.
An air-filled cubical cavity of size 3.2 cm is made of brass (σc = 1.37 X 107 S/m). Calculate: (a) The resonant frequency of the TE101 mode, (b) The quality factor at that mode.
Design an air-filled cubical cavity to have its dominant resonant frequency at 3 GHz.
An air-filled cubical cavity of size 10 cm hasE = 200 sin 30πx sin 30πy cos 6 X 109t az, V/mFind H.
Use Gauss's theorem [and A.21) if necessary] to prove the following:(a) Any excess charge placed on a conductor must lie entirely on its surface. (A conductor by definition contains charges capable
Using Dirac delta functions in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities p(x).(a) In spherical coordinates, a charge Q uniformly
Each of three charged spheres of radius a, one conducting, one having a uniform charge density within its volume, and one having a spherically symmetric charge density that varies radially as rn (n
The time-averaged potential of a neutral hydrogen atom is given by where q is the magnitude of the electronic charge, and ??1 = ?0/2, ?0 being the Bohr radius. Find the distribution of charge
A simple capacitor is a device formed by two insulated conductors adjacent to each other. If equal and opposite charges are placed on the conductors, there will be a certain difference of potential
Two long, cylindrical conductors of radii a1 and a2 are parallel and separated by a distance d, which is large compared with either radius. Show that the capacitance per unit length is given
(a) For the three capacitor geometries in Problem 1.6 calculate the total electrostatic energy and express it alternatively in terms of the equal and opposite charges Q and –Q placed on the
Calculate the attractive force between conductors in the parallel plate capacitor (Problem 1.6a) and the parallel cylinder capacitor (Problem 1.7) for(a) Fixed charges on each conductor;(b) Fixed
Prove the mean value theorem: For charge-free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered on that
Use Gauss's theorem to prove that at the surface of a curved charged conductor, the normal derivative of the electric field is given by where R1 and R2 are the principal radii of curvature of
Prove Green's reciprocation theorem: If ? is the potential due to a volume-charge density ? within a volume V and a surface-charge density a on the conducting surface S bounding the volume V, while
Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use the reciprocation theorem of Green to prove that the total induced
Consider the electrostatic Green functions of Section 1.10 for Dirichlet and Neumann boundary conditions on the surface S bounding the volume V. Apply Green's theorem (1.35) with integration variable
Prove Thomson's theorem: If a number of surfaces are fixed in position and a given total charge is placed on each surface, then the electrostatic energy in the region bounded by the surfaces is an
A volume V in vacuum is bounded by a surface S consisting of several separate conducting surfaces Si. One conductor is held at unit potential and all the other conductors at zero potential.(a) Show
Consider the configuration of conductors of Problem 1.17, with all conductors except S1 held at zero potential. (a) Show that the potential ?(?) anywhere in the volume V and on any of the surfaces Si
For the cylindrical capacitor of Problem 1.6c, evaluate the variational upper bound of Problem 1.17b with the naive trial function, ψ1(ρ) = (b – ρ)/(b – a). Compare the variational result with
In estimating the capacitance of a given configuration of conductors, comparison with known capacitances is often helpful. Consider two configurations of n conductors in which the (n – 1)
A point charge q is brought to a position a distance d away from an infinite plane conductor held at zero potential. Using the method of images, find: (a) The surface-charge density induced on the
Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting sphere of inner radius a. Find (a) The potential inside the sphere; (b) The induced
A straight-line charge with constant linear charge density ? is located perpendicular to the x-y plane in the first quadrant at (x0, y0). The intersecting planes x = 0, ? ? 0 and ? = 0, x ? 0 are
A point charge is placed a distance d > R from the center of an equally charged, isolated, conducting sphere of radius R.a) Inside of what distance from the surface of sphere in the point charge
(a) Show that the work done to remove the charge q from a distance r > a to infinity against the force, Eq. (2.6), of a grounded conducting sphere is Relate this result to the electrostatic
The electrostatic problem of a point charge q outside an isolated, charged conducting sphere is equivalent to that of three charges, the original and two others, one located at the center of the
Consider a potential problem in the half-space defined by z ? 0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). (a) Write down the appropriate Green function G(x, x'). (b)
A two-dimensional potential problem is defined by two straight parallel line charges separated by a distance R with equal and opposite linear charge densities ? and ? ?. (a) Show by direct
An insulated, spherical, conducting shell of radius a is in a uniform electric field E0. If the sphere is cut into two hemispheres by a plane perpendicular to the field, find the force required to
A large parallel plate capacitor is made up of two plane conducting sheets with separation D, one of which has a small hemispherical boss of radius a on its inner surface (D >> a). The
A line charge with linear charge density τ is placed parallel to, and a distance R away from, the axis of a conducting cylinder of radius b held at fixed voltage such that the potential vanishes at
Starting with the series solution (2.71) for the two-dimensional potential problem with the potential specified on the surface of a cylinder of radius b, evaluate the coefficients formally,
(a) Two halves of a long hollow conducting cylinder of inner radius b are separated by small lengthwise gaps on each side, and are kept at different potentials Vl and V2. Show that the potential
A variant of the preceding two-dimensional problem is a long hollow conducting cylinder of radius b that is divided into equal quarters, alternate segments being held at potential + V and ? V. (a)
(a) Show that the Green function G(x, y; x', y?) appropriate for Dirichlet boundary conditions for a square two-dimensional region, 0 ? x ? l, 0 ? y ? l, has an expansion Where gn(y, ?y?)
A two-dimensional potential exists on a unit square area (0 ? x ? l, 0 ? y ? l) bounded by "surfaces" held at zero potential. Over the entire square there is a uniform charge density of unit strength
(a) Construct the free-space Green function G(x, ?; ??, ?') for two-dimensional electrostatics by integrating 1/R with respect to (z? - z) between the limits ?Z, where Z is taken to be very large.
(a) By finding appropriate solutions of the radial equation in part b of Problem 2.17, find the Green function for the interior Dirichlet problem of a cylinder of radius b [gm(?, ?' = b) = 0. See
Use Cauchy's theorem to derive the Poisson integral solution. Cauchy's theorem states that if F(z) is analytic in a region R bounded by a closed curve C,then
(a) For the example of oppositely charged conducting hemispherical shells separated by a tiny gap, as shown in Figure 2.8, show that the interior potential (r Find the first few terms of the
A hollow cube has conducting walls defined by six planes x = 0, у = 0, z = 0, and x = а, у = a, z = a. The walls z = 0 and z = a are held at a constant potential V. The other four sides are at
In the two-dimensional region shown in Fig, the angular functions appropriate for Dirichlet boundary conditions at ? = 0 and ? = ? are ?(?) = Am sin(m??/?). Show that the completeness relation for
A closed volume is bounded by conducting surfaces that are the n sides of a regular polyhedron (n = 4, 6, 8, 12, 20). The n surfaces are at different potentials Vi, i = 1, 2,..., n. Prove in the
Using the results of Problem 2.29, apply the Galerkin method to the integral equivalent of the Poisson equation with zero potential on the boundary, For the lattice of Problem 1.24, with its three
A spherical surface of radius R has charge uniformly distributed over its surface with a density Q/4?R2, except for a spherical cap at the north pole, defined by the cone ? = ?. (a) Show that the
A thin, flat, conducting, circular disc of radius R is located in the x-y plane with its center at the origin, and is maintained at a fixed potential V. With the information that the charge density
A hollow sphere of inner radius a has the potential specified on its surface to be Ф = V(θ, ф). Prove the equivalence of the two forms of solution for the potential inside the sphere:
Two point charges q and –q are located on the z axis at z = + a and z = –a, respectively.(a) Find the electrostatic potential as an expansion in spherical harmonics and powers of r for both r
Three point charges (q, ?2q, q) are located in a straight line with separation a and with the middle charge (?2q) at the origin of a grounded conducting spherical shell of radius b, as indicated in
A hollow right circular cylinder of radius b has its axis coincident with the z axis and its ends at z = 0 and z = L. The potential on the end faces is zero, while the potential on the cylindrical
For the cylinder in Problem 3.9 the cylindrical surface is made of two equal half-cylinders, one at potential V and the other at potential ? V, so that (a) Find the potential inside the
A modified Bessel-Fourier series on the interval 0 ? ? ? a for an arbitrary function f?(?) can be based on the "homogeneous" boundary conditions: The first condition restricts v. The second
An infinite, thin, plane sheet of conducting material has a circular hole of radius a cut in it. A thin, flat disc of the same material and slightly smaller radius lies in the plane, filling the
Solve for the potential in Problem 3.1, using the appropriate Green function obtained in the text, and verify that the answer obtained in this way agrees with the direct solution from the
A line charge of length 2d with a total charge Q has a linear charge density varying as (d2 – z2), where z is the distance from the midpoint. A grounded, conducting, spherical shell of inner radius
The Dirichlet Green function for the unbounded space between the planes at z = 0 and z = L allows discussion of a point charge or a distribution of charge between parallel conducting planes held at
Consider a point charge q between two infinite parallel conducting planes held at zero potential. Let the planes be located at z = 0 and z = L in a cylindrical coordinate system, with the charge on

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