Questions and Answers of Electrodynamics

Transform the following vectors to cylindrical and spherical coordinates:(a) D = (x + z)ay(b) E = (y2 - x2)ax + xyzay + (x2 - z2)az
Convert the following vectors to cylindrical and spherical systems:
Express the following vectors in Cartesian coordinates: (a) A = p(z2 + l)ap - pz cos Φ aΦ (b) B = 2r sin θ cos Φ ar + r cos θ cos θ aθ – r sin Φ aΦ
Convert the following vectors to Cartesian coordinates:
(a) Express the vector field in cylindrical and spherical coordinates,H = xy2zax + x2yzay + xyz2az(b) In both cylindrical and spherical coordinates, determine H at (3, – 4, 5).
Let A = p cos θ ap + pz2 sin θ az (a) Transform A into rectangular coordinates and calculate its magnitude at point (3, – 4, 0). (b) Transform A into spherical system and calculate its
The transformation (Ap, AΦ, Az) → (Ax, Ay, Az) in eq. (2.15) is not complete. Complete it by expressing cos Φ and sin Φ in terms of x, y, and z. Do the same thing to the
In Practice Exercise 2.2, express A in spherical and B in cylindrical coordinates. Evaluate A at (10, π/2, 3π/4) and B at (2, π/6, 1).
Calculate the distance between the following pairs of points: (a) (2, 1, 5) and (6, – 1, 2) (b) (3, π/2, – 1) and (5, 3 π/2, 5) (c) (10, π/4, 3π/4) and (5, π/6,
Describe the intersection of the following surfaces: (a) x = 2, y = 5 (b) x = 2, y = – 1, z = 10 (c) r = 10, θ = 30o (d) ρ = 5, Φ = 40o (e) Φ = 60o, z = 10 (f) r =
At point T(2, 3, –4), express az in the spherical system and ar in the rectangular system.
Given vectors A = 2ax + 4ay + 10az and B = – 5ap + aΦ – 3az, find (a) A + B at P(0, 2, – 5 ) (b) The angle between A and B at P (c) The scalar component of A along B at P
Given that G = (x + y2)ax + xzay + (z2 + zy)az, find the vector component of G along aΦ at point P(8, 30°, 60°). Your answer should be left in the Cartesian system.
If J = r sin θ cos Φ ar – cos 2θ sin Φ aθ + tan θ/2 In r aΦ at T(2, π/2, 3π/2), determine the vector component of J that is (a) Parallel to az (b)
Let H – 5p sin Φ ap – pz cos Φ aθ + 2paz. At point P(2, 30°, – 1), find:(a) A unit vector along H(b) The component of H parallel to ax(c) The component of H normal to p = 2(d) The component
Another way of defining a point P in space is (r, a, ?, ?) where the variables are portrayed in Figure. Using this definition, find (r, a, ?, ?) for the following points:(a) (??2, 3, 6)(b) (4, 30?,
A vector field in "mixed" coordinate variables is given byExpress G completely in spherical system
A half-wave dipole is made of copper and is of diameter 2.6 mm. Determine the efficiency of the dipole if it operates at 15 MHz.
Find Uave, Umax, and D if: (a) U(θ, Φ) = sin2 2θ, 0 < 0 < π, 0 < 0 < 2π (b) U(θ, Φ) = 4 esc2 2θ, π/3 < 0 < π/2, 0 < Φ < π (c) U(θ,
For the following radiation intensities, find the directive gain and directivity: (a) U(θ, Φ)= sin2 θ, 0 < θ < π, 0 < Φ < 2π (b) U(θ, Φ)= 4 sin2 θ
In free space, an antenna radiates a field At far field. Determine: (a) The total radiated power, v (b) The directive gain at ? = 60°
Derive Es at far field due to the two-element array shown in Figure. Assume that the Hertzian dipole elements are fed in phase with uniform current Io cos ?t.
An array comprises two dipoles that are separated by one wavelength. If the dipoles are fed by currents of the same magnitude and phase,(a) Find the array factor.(b) Calculate the angles where the
An array of two elements that are fed by currents that are 180° out of phase with each other. Plot the group pattern if the elements are separated by:(a) d = λ/4,(b) d = λ/2
Sketch the group pattern in the xz-plane of the two-element array of Figure with(a) d = ?, ? = ?/2(b) d = ?/4, ? = 3?/4(c) d = 3A/4, ? = 0
An antenna array consists of N identical Hertzian dipoles uniformly located along the z-axis and polarized in the z-direction. If the spacing between the dipole is λ/4, sketch the group pattern
Sketch the resultant group patterns for the four-element arrays shown inFigure.
For a 10-turn loop antenna of radius 15 cm operating at 100 MHz, calculate the effective area at θ = 30°, Φ = 90°.
An antenna receives a power of 2 μW from a radio station. Calculate its effective area if the antenna is located in the far zone of the station where E = 50 mV/m.
(a) Show that the Friis transmission equation can be written as(b) Two half-wave dipole antennas are operated at 100 MHz and separated by 1 km. If 80 W is transmitted by one, how much power is
The electric field strength impressed on a half-wave dipole is 3 mV/m at 60 MHz. Calculate the maximum power received by the antenna. Take the directivity of the half-wave dipole as 1.64.
The power transmitted by a synchronous orbit satellite antenna is 320 W. If the antenna has a gain of 40 dB at 15 GHz, calculate the power received by another antenna with a gain of 32 dB at the
The directive gain of an antenna is 34 dB. If the antenna radiates 7.5 kW at a distance of 40 km, find the time-average power density at that distance.
Two identical antennas in an anechoic chamber are separated by 12 m and are oriented for maximum directive gain. At a frequency of 5 GHz, the power received by one is 30 dB down from that transmitted
What is the maximum power that can be received over a distance of 1.5 km in free space with a 1.5-GHz circuit consisting of a transmitting antenna with a gain of 25 dB and a receiving antenna with a
An L-band pulse radar with a common transmitting and receiving antenna having a directive gain of 3500 operates at 1500 MHz and transmits 200 kW. If the object is 120 km from the radar and its
A transmitting antenna with a 600 MHz carrier frequency produces 80 W of power. Find the power received by another antenna at a free space distance of 1 km. Assume both antennas has unity power gain.
A monostable radar operating at 6 GHz tracks a 0.8 m2 target at a range of 250 m. If the gain is 40 dB, calculate the minimum transmitted power that will give a return power of 2μW.
In the bistatic radar system of Figure, the ground-based antennas are separated by 4 km and the 2.4 m2 target is at a height of 3 km. The system operates at 5 GHz. For Gdt of 36 dB and Gdr of 20 dB,
Discuss briefly some applications of microwaves other than those discussed in the text.
A useful set of parameters, known as the scattering transfer parameters, is related to the incident and reflected waves as(a) Express the T-parameters in terms of the S-parameters.(b) Find T when
The S-parameters of a two-port network are: S11 = 0.33 – j0.16, Sl2 = S21 = 0.56, S22 = 0.44 – j0.62 Find the input and output reflection coefficients when ZL = Zo = 50 Ω and Zg = 2Zo.
Why can't regular lumped circuit components such as resistors, inductors, and capacitors be used at microwave frequencies?
An electrostatic discharge (ESD) can be modeled as a capacitance of 125 pF charged to 1500 V and discharging through a 2-km resistor. Obtain the current waveform.
The insertion loss of a filter circuit can be calculated in terms of its A, B, C and D parameters when terminated by Zg and ZL as shown in Figure. Show that
A silver rod has rectangular cross section with height 0.8 cm and width 1.2 cm. Find:(a) The dc resistance per 1 km of the conductor(b) The ac resistance per 1 km of the conductor at 6 MHz
A glass fiber has a core diameter of 50 μm, a core refractive index of 1.62, and a cladding with a refractive index of 1.604. If light having a wavelength of 1300 nm is used, find:(a) The numerical
An optical fiber with a radius of 2.5 μm and a refractive index of 1.45 is surrounded by an air cladding. If the fiber is illuminated by a ray of 1.3 μm light, determine: (a) V (b)
An optical fiber with an attenuation of 0.4 dB/km is 5 km long. The fiber has n1 = 1.53,n2 = 1.45, and a diameter of 50 μm. Find:(a) The maximum angle at which rays will enter the fiber and be
Attenuation α10 in Chapter 10 is in Np/m, whereas attenuation αl4 in this chapter is in dB/km. What is the relationship between the two?
Given the one-dimensional differential equationSubject to y(0) = 0, y(l) = 10, use the finite difference (iterative) method to find y(0.25). You may take A = 0.25 and perform 5iterations.
(a) From the table below, obtain dV/dx and, d2x/dx2 at x = 0.15.(b) The data in the table above are obtained from V = 10 sin x. Compare your result in part (a) with the exactvalues.
Show that the finite difference equation for Laplace's equation in cylindrical coordinates, V = V(p, z), is Where h = ?z ? ??.
Using the finite difference representation in cylindrical coordinates (?, ?) at a grid point P shown in Figure, let ? = m?? and ? = n ?? so that V(?, ?)|? = V(m??, n??) = Vnm. Show that
Use FDM to calculate the potentials at nodes 1 and 2 in the potential system shown in figure
Rework Problem 15.7 if ρs = 100/π nC/m2, h = 0.l m, and ε = εo, where h is the mesh size.Rework Problem 15.7Use FDM to calculate the potentials at nodes 1 and 2 in the potential system
Consider the potential system shown in Figure.(a) Set the initial values at the free nodes equal to zero and calculate the potential at the free nodes for five iterations,(b) Solve the problem by the
Apply the band matrix technique to set up a system of simultaneous difference equations for each of the problems in Figure. Obtain matrices [A] and[B].
(a) How would you modify matrices [A]?and [B]?of Example 15.3 if the solution region had charge density ?s? (b) Write a program to solve for the potentials at the grid points shown in Figure assuming
The two-dimensional wave equation is given byBy letting ?jm,n denote the finite difference approximation of ?(xm, zn, tj), show that the finite difference scheme for the wave equation isWhere h = ?x
Write a program that uses the finite difference scheme to solve the one-dimensional wave equationGiven boundary conditions V(0, i) = 0, V(l, t) = 0, t > 0 and the initial condition ?V/?t (x, 0) =
Show that the finite difference representation of Laplace's equation using the ninenode molecule of Figure isVo = 1/8 (V1 + V2 + V3 + V4 + V5 + V6 + V7 +V8)
A transmission line consists of two identical wires of radius a,?separated by distance d?as shown in Figure. Maintain one wire at 1 V and the other at ? 1 V and use MOM to find the capacitance per
Determine the potential and electric field at point (–1, 4, 5) due to the thin conducting wire of Figure. Take Vo = 1 V, L = 1 m, a = 1 mm.
Two conducting wires of equal length L?and radius a?are separated by a small gap and inclined at an angle ??as shown in Figure. Find the capacitance between the wires using the method of moments for
Consider the coaxial line of arbitrary cross section shown in Figure (a). Using the moment method to find the capacitance C per length involves dividing each conductor into N strips so that the
The conducting bar of rectangular cross section is shown in Figure. By dividing the bar into Nequal segments, we obtain the potential at thejth segment asWhereAnd ? is the length of the segment. If
Another way of defining the shape functions at an arbitrary point (x, y) in a finite element is using the areas A1,A2, and A3 shown in Figure. Show that?k = Ak/A, k = 1, 2, 3Where A = A, + A2 + A3 is
For each of the triangular elements of Figure,(a) Calculate the shape functions.(b) Determine the coefficientmatrix.
The nodal potential values for the triangular element of Figure are V1 = 100 V, V2 = 50 V, and V3 = 30 V.(a) Determine where the 80 V equipotential line intersects the boundaries of the element,(b)
The triangular element shown in Figure is part of a finite element mesh. If V1 = 8 V, V2 = 12 V, and V3 = 10 V, find the potential at (a) (1, 2) and (b) The center of theelement.
Determine the global coefficient matrix for the two-element region shown inFigure.
Find the global coefficient matrix of the two-element mesh ofFigure.
For the two-element mesh of Figure, let Vi = 10 V and V3 = 30 V. Find V2 and V4.
The mesh in Figure is part of a large mesh. The shading region is conducting and has no elements. Find C5,5 andC5,1.
Use the program to solve Laplace's equation in the problem shown in Figure where Vo = 100 V. Compare the finite element solution to the exact solution in Example 6.5; thatis,
Repeat the preceding problem for Vo = 100 sin ?x. Compare the finite element solution with the theoretical solution [similar to Example 6.6(a)]; that is,
Show that when a square mesh is used in FDM, we obtain the same result in FEM when the squares are cut into triangles
Point charges Q1 = 5µC and Q2 = – 4µC are placed at (3, 2, 1) and (–4, 0, 6), respectively. Determine the force on Q1.
Five identical 15-µC point charges are located at the center and corners of a square defined by –1 < x, y < 1, z = 0.(a) Find the force on the 10-µC point charge at (0, 0, 2).(b) Calculate the
Point charges Q1 and Q2 are, respectively, located at (4, 0, – 3) and (2, 0, 1). If Q2 = 4nC, find Q1 such that(a) The E at (5, 0, 6) has no z-component(b) The force on a test charge at (5, 0, 6)
Charges + Q and + 3Q are separated by a distance 2 m. A third charge is located such that the electrostatic system is in equilibrium. Find the location and the value of the third charge in terms of Q.
Determine the total charge
Calculate the total charge due to the charge distributions labeled A, B, C infigure.
Find E at (5, 0, 0) due to charge distribution labeled A infigure.
Due to the charge distribution labeled B in figure,(a) Find E at point (0, 0, 3) if ?S = 5 mC/m2.(b) Find E at point (0, 0, 3) if ?S = 5 sin ? mC/m2.
A circular disk of radius a carries charge ρs = 1/ρ C/m2. Calculate the potential at (0, 0, h).
A ring placed along y2 + z2 = 4, x = 0 carries a uniform charge of 5µC/m.(a) Find D at P(3,0, 0).(b) If two identical point charges Q are placed at (0, – 3, 0) and (0, 3, 0) in addition to the
(a) Show that the electric field at point (0, 0, h) due to the rectangle described by ? a x a, ? b y b, z = 0 carrying uniform charge ?sC/m2 is (b) If a = 2, b?= 5, ?s =?10?5, find the total
A point charge 100pC is located at (4, 1, – 3) while the x-axis carries charge 2nC/m. If the plane z = 3 also carries charge 5 nC/m2, find E at (1, 1, 1).
Line x = 3, z = – 1 carries charge 20nC/m while plane x = –2 carries charge 4 nC/m2. Find the force on a point charge – 5mC located at the origin.
Point charges are placed at the corners of a square of size 4 m as shown in figure. If Q?= 15?C, find D at (0, 0, 6).
State Gauss's law. Deduce Coulomb's law from Gauss's law thereby affirming that Gauss's law is an alternative statement of Coulomb's law and that Coulomb's law is implicit in Maxwell's equation
Determine the charge density due to each of the following electric fluxdensities:
Let E = xyax + x2ay, find (a) Electric flux density D (b) The volume charge density ρv.
Plane x + 2y = 5 carries charge ρS = 6nC/m2. Determining E at (– 1, 0, 1)
In free space, D = 2v2ax, t + 4xy - az mC/m2. Find the total charge stored in the region 1 < x < 2, 1 < y < 2, – 1 < z < 4.
In a certain region, the electric field is given by D = 2ρ(z + l)cos Φ aρ – ρ(z + l)sin Φ aΦ + ρ2 cos Φ az µC/m2 (a) Find the charge density. (b)
The Thomson model of a hydrogen atom is a sphere of positive charge with an electron (a point charge) at its center. The total positive charge equals the electronic charge e. Prove that when the
Three concentric spherical shells r = 1, r = 2, and r = 3 m, respectively, have charge distributions 2, - 4, and 5 µC/m2.(a) Calculate the flux through r = 1.5 m and r = 2.5 m.(b) Find D at r = 0.5,

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