Questions and Answers of Principles Communications Systems

A point charge, Q = −0.3μC and m = 3 × 10−16 kg, is moving through the field E = 30az V/m. Use Eq. (1) and Newton’s laws to develop the appropriate differential equations and solve them,
By expanding Eq. (58), Section 7.7 in rectangular coordinates, show that (59) is correct.Eq. (58)∇×∇×A ≡ ∇(∇ · A)−∇2A
Compute the vector magnetic potential within the outer conductor for the coaxial line whose vector magnetic potential is shown in Figure 7.20 if the outer radius of the outer conductor is 7a. Select
Show that ∇2(1/R12) = −∇1(1/R12) = R21/R312.
Assume that A = 50ρ2az Wb/m in a certain region of free space.(a) Find H and B.(b) Find J.(c) Use J to find the total current crossing the surface 0 ≤ ρ ≤ 1, 0 ≤ ϕ < 2π, z = 0.(d) Use
A current sheet, K = 20 az A/m, is located at ρ = 2, and a second sheet, K = −10az A/m, is located at ρ = 4.(a) Let Vm = 0 at P(ρ = 3, ϕ = 0, z = 5) and place a barrier at ϕ = π. Find Vm(ρ,
A solid cylinder of radius a and length L, where L >>a, contains volume charge of uniform density ρ0 C/m3. The cylinder rotates about its axis (the z axis) at angular velocity Ω rad/s.(a)
In Figure 7.22, let the regions 0 < z < 0.3 m and 0.7 < z < 1.0 m be conducting slabs carrying uniform current densities of 10 A/m2in opposite directions as shown. Find H at z =:Figure
An infinite filament on the z axis carries 20π mA in the az direction. Three az-directed uniform cylindrical current sheets are also present: 400 mA/m at ρ = 1 cm, −250 mA/m at ρ = 2 cm, and
A hollow spherical conducting shell of radius a has filamentary connections made at the top (r = a, θ = 0) and bottom (r = a, θ = π). A direct current I flows down the upper filament, down the
A current sheet K = 8ax A/m flows in the region −2 < y < 2 in the plane z = 0. Calculate H at P(0, 0, 3).
For the finite-length current element on the z axis, as shown in Figure 7.5, use the Biot-Savart law to derive Eq. (9) of Section 7.1.Eq. (9)Figure 7.5 (sin a2 – sina1)as 4лр н Point 2
A filamentary conductor carrying current I in the az direction extends along the entire negative z axis. At z = 0 it connects to a copper sheet that fills the x > 0, y > 0 quadrant of the xy
A disk of radius a lies in the xy plane, with the z axis through its center. Surface charge of uniform density ρs lies on the disk, which rotates about the z axis at angular velocity Ω rad/s. Find
The parallel filamentary conductors shown in Figure 7.21 lie in free space. Plot |H| versus y,ˆ’4 < y < 4, along the line x = 0, z = 2.Figure 7.21 (0, -1, 0) (0, 1, 0) 1A/ 1A
Two circular current loops are centered on the z axis at z = ± h. Each loop has radius a and carries current I in the aϕ direction.(a) Find H on the z axis over the range −h < z < h. Take I
(a) Find H in rectangular components at P(2, 3, 4) if there is a current filament on the z axis carrying 8 mA in the az direction.(b) Repeat if the filament is located at x = −1, y = 2.(c) Find H
In free space, let ρν = 2000/r2.4.(a) Use Poisson’s equation to find V(r) if it is assumed that r2Er → 0 when r → 0, and also that V → 0 as r → ∞. (b) Now find V(r) by using
A potential field in free space is given as V = 100 ln tan(θ/2) + 50 V.(a) Find the maximum value of |Eθ| on the surface θ = 40◦ for 0.1 < r < 0.8 m, 60◦ < ϕ < 90◦.(b) Describe
Two coaxial conducting cones have their vertices at the origin and the z axis as their axis. Cone A has the point A(1, 0, 2) on its surface, while cone B has the point B(0, 3, 2) on its surface. Let
The hemisphere 0 < r < a, 0 < θ < π/2, is composed of homogeneous conducting material of conductivity σ. The flat side of the hemisphere rests on a perfectly conducting plane. Now, the
Concentric conducting spheres are located at r = 5 mm and r = 20 mm. The region between the spheres is filled with a perfect dielectric. If the inner sphere is at 100 V and the outer sphere is at 0
Repeat Problem 6.37, but with the dielectric only partially filling the volume, within 0 < ϕ < π, and with free space in the remaining volume.In ProblemCoaxial conducting cylinders are
A parallel-plate capacitor is made using two circular plates of radius a, with the bottom plate on the xy plane, centered at the origin. The top plate is located at z = d, with its center on the z
The conducting planes 2x + 3y = 12 and 2x + 3y = 18 are at potentials of 100 V and 0, respectively. Let ∈ = ∈0 and find(a) V at P(5, 2, 6);(b) E at P.
Consider the parallel-plate capacitor of Problem 6.30, but this time the charged dielectric exists only between z = 0 and z = b, where b < d. Free space fills the region b < z < d. Both
A uniform volume charge has constant density ρν = ρ0 C/m3 and fills the region r < a, in which permittivity is assumed. A conducting spherical shell is located at r = a and is held at ground
Let V = (cos 2ϕ)/ρ in free space.(a) Find the volume charge density at point A(0.5, 60◦, 1).(b) Find the surface charge density on a conductor surface passing through the point B(2, 30◦, 1).
A parallel-plate capacitor has plates located at z = 0 and z = d. The region between plates is filled with a material that contains volume charge of uniform density ρ0 C/m3 and has permittivity ∈.
Given the potential field V = (Aρ4 + Bρ−4) sin 4ϕ:(a) Show that ∇2V = 0.(b) Select A and B so that V = 100 V and |E| = 500 V/m at P(ρ = 1, ϕ = 22.5◦, z = 2).
The inner conductor of the transmission line shown in Figure 6.13 has a square cross section 2a Ã? 2a, whereas the outer square is 4a Ã? 5a. The axes are displaced as shown. (a) Construct a
A solid conducting cylinder of 4 cm radius is centered within a rectangular conducting cylinder with a 12 cm by 20 cm cross section.(a) Make a full-size sketch of one quadrant of this configuration
Construct a curvilinear-square map of the potential field between two parallel circular cylinders, one of 4 cm radius inside another of 8 cm radius. The two axes are displaced by 2.5 cm. These
Construct a curvilinear-square map of the potential field about two parallel circular cylinders, each of 2.5 cm radius, separated by a centerto-center distance of 13 cm. These dimensions are suitable
Construct a curvilinear-square map for a coaxial capacitor of 3 cm inner radius and 8 cm outer radius. These dimensions are suitable for the drawing.(a) Use your sketch to calculate the capacitance
Consider an arrangement of two isolated conducting surfaces of any shape that form a capacitor. Use the definitions of capacitance (Eq. (2) in this chapter) and resistance (Eq. (14) in Chapter 5) to
A 2-cm-diameter conductor is suspended in air with its axis 5 cm from a conducting plane. Let the potential of the cylinder be 100 V and that of the plane be 0 V.(a) Find the surface charge density
Two #16 copper conductors (1.29 mm diameter) are parallel with a separation d between axes. Determine d so that the capacitance between wires in air is 30 pF/m.
With reference to Figure 6.5, let b = 6 m, h = 15 m, and the conductor potential be 250 V. Take ∈ = 0. Find values for K1, ρL, a, and C.
(a) Determine the capacitance of an isolated conducting sphere of radius a in free space (consider an outer conductor existing at r → ∞).(b) The sphere is to be covered with a dielectric layer of
Two conducting spherical shells have radii a = 3 cm and b = 6 cm. The interior is a perfect dielectric for which r = 8.(a) Find C.(b) A portion of the dielectric is now removed so that ∈r = 1.0, 0
A coaxial cable has conductor dimensions of a = 1.0 mm and b = 2.7 mm. The inner conductor is supported by dielectric spacers (∈r = 5) in the form of washers with a hole radius of 1 mm and an outer
Two coaxial conducting cylinders of radius 2 cm and 4 cm have a length of 1 m. The region between the cylinders contains a layer of dielectric from ρ = c to ρ = d with r = 4. Find the capacitance
A parallel-plate capacitor is made using two circular plates of radius a, with the bottom plate on the xy plane, centered at the origin. The top plate is located at z = d, with its center on the z
Let ∈r1 = 2.5 for 0 < y < 1 mm, ∈r2 = 4 for 1 < y < 3 mm, and ∈r3 for 3 < y < 5 mm (region 3). Conducting surfaces are present at y = 0 and y = 5 mm. Calculate the capacitance
Repeat Problem 6.4, assuming the battery is disconnected before the plate separation is increased.In ProblemAn air-filled parallel-plate capacitor with plate separation d and plate area A is
Consider a composite material made up of two species, having number densities N1 and N2 molecules/m3, respectively. The two materials are uniformly mixed, yielding a total number density of N = N1 +
A coaxial conductor has radii a = 0.8 mm and b = 3 mm and a polystyrene dielectric for which ∈r = 2.56. If P = (2/ρ)aρ nC/m2 in the dielectric, find(a) D and E as functions of ρ;(b) Vab and
Atomic hydrogen contains 5.5 × 1025 atoms/m3 at a certain temperature and pressure. When an electric field of 4 kV/m is applied, each dipole formed by the electron and positive nucleus has an
The line segment x = 0,−1 ≤ y ≤ 1, z = 1, carries a linear charge density ρL = π|y|μ C/m. Let z = 0 be a conducting plane and determine the surface charge density at:(a) (0, 0, 0);(b) (0, 1,
In cylindrical coordinates, let ρν = 0 forρ < 1 mm, ρν = 2 sin(2000 πρ) nC/m3 for 1 mm < ρ < 1.5 mm, and ρν = 0 for ρ > 1.5 mm. Find D everywhere.
A crude device for measuring charge consists of two small insulating spheres of radius a, one of which is fixed in position. The other is movable along the x axis and is subject to a restraining
A hollow cylindrical shell of radius a is centered on the z axis and carries a uniform surface current density of Kaaφ.(a) Show that H is not a function of ϕ or z.(b) Show that Hϕ and Hρ are
A toroid having a cross section of rectangular shape is defined by the following surfaces: the cylinders ρ = 2 and ρ = 3 cm, and the planes z = 1 and z = 2.5 cm. The toroid carries a surface
Assume that there is a region with cylindrical symmetry in which the conductivity is given by σ = 1.5e−150ρ kS/m. An electric field of 30az V/m is present.(a) Find J.(b) Find the total current
A current filament carrying I in the −az direction lies along the entire positive z axis. At the origin, it connects to a conducting sheet that forms the xy plane.(a) Find K in the conducting
A current filament on the z axis carries a current of 7 mA in the az direction, and current sheets of 0.5 az A/m and −0.2 az A/m are located at ρ = 1 cm and ρ = 0.5 cm, respectively. Calculate H
A wire of 3 mm radius is made up of an inner material (0 < ρ < 2 mm) for which σ = 107 S/m, and an outer material (2 mm < ρ < 3 mm) for which σ = 4×107 S/m. If the wire carries a
In spherical coordinates, the surface of a solid conducting cone is described by θ = π/4 and a conducting plane by θ = π/2. Each carries a total current I. The current flows as a surface current
A solid conductor of circular cross section with a radius of 5 mm has a conductivity that varies with radius. The conductor is 20 m long, and there is a potential difference of 0.1 V dc between its
A cylindrical wire of radius a is oriented with the z axis down its center line. The wire carries a nonuniform current down its length of density J = bρ az A/m2 where b is a constant.(a) What total
Given the field H = 20ρ2aϕ A/m:(a) Determine the current density J.(b) Integrate J over the circular surface ρ ≤ 1, 0 < ϕ < 2π, z = 0, to determine the total current passing through that
Infinitely long filamentary conductors are located in the y = 0 plane at x = n meters where n = 0, ±1, ±2, . . . Each carries 1 A in the azdirection.(a) Find H on the y axis. As a
When x, y, and z are positive and less than 5, a certain magnetic field intensity may be expressed as H = [x2yz/(y + 1)]ax + 3x2z2ay − [xyz2/(y + 1)]az. Find the total current in the ax direction
Consider a sphere of radius r = 4 centered at (0, 0, 3). Let S1 be that portion of the spherical surface that lies above the xy plane. Find ʃS1 (∇ × H) · dS if H = 3ρ aϕ in cylindrical
The magnetic field intensity is given in a certain region of space as H = [(x + 2y)/z2]ay + (2/z)az A/m.(a) Find ∇× H.(b) Find J.(c) Use J to find the total current passing through the surface z =
Given H = (3r2/ sin θ)aθ + 54r cos θaϕ A/m in free space:(a) Find the total current in the aθ direction through the conical surface θ = 20◦, 0 ≤ ϕ ≤ 2π, 0 ≤ r ≤ 5, by whatever side
A long, straight, nonmagnetic conductor of 0.2 mm radius carries a uniformly distributed current of 2 A dc.(a) Find J within the conductor.(b) Use Ampere’s circuital law to find H and B within the
A solid, nonmagnetic conductor of circular cross section has a radius of 2 mm. The conductor is in homogeneous, with σ = 106(1 + 106ρ2) S/m. If the conductor is 1 m in length and has a voltage of 1
The cylindrical shell defined by 1 cm < ρ < 1.4 cm consists of a nonmagnetic conducting material and carries a total current of 50 A in the az direction. Find the total magnetic flux crossing
The free space region defined by 1 < z < 4 cm and 2 < ρ < 3 cm is a toroid of rectangular cross section. Let the surface at ρ = 3 cm carry a surface current K = 2az kA/m.(a) Specify the
Use an expansion in rectangular coordinates to show that the curl of the gradient of any scalar field G is identically equal to zero.
A filamentary conductor on the z axis carries a current of 16 A in the az direction, a conducting shell at ρ = 6 carries a total current of 12 A in the −az direction, and another shell at ρ = 10
Let A = (3y − z)ax + 2xzay Wb/m in a certain region of free space.(a) Show that ∇ · A = 0.(b) At P(2,−1, 3), find A, B, H, and J.
Let N = 1000, I = 0.8 A, Ï0= 2 cm, and a = 0.8 cm for the toroid shown in Figure 7.12b. Find Vmin the interior of the toroid if Vm= 0 at Ï = 2.5 cm, Ï• =
A square filamentary differential current loop, dL on a side, is centered at the origin in the z = 0 plane in free space. The current I flows generally in the aÏ•direction.(a) Assuming that
Planar current sheets of K = 30az A/m and −30az A/m are located in free space at x = 0.2 and x = −0.2, respectively. For the region −0.2 < x < 0.2(a) Find H;(b) Obtain an expression for
Show that the line integral of the vector potential A about any closed path is equal to the magnetic flux enclosed by the path, or ∮ A· dL = ʃ B · dS.
A half-wave dipole antenna is known to have a maximum effective area, given as Amax.(a) Write the maximum directivity of this antenna in terms of Amax and wavelength λ.(b) Express the current
Signals are transmitted at a 1-m carrier wavelength between two identical half-wave dipole antennas spaced by 1 km. The antennas are oriented such that they are exactly parallel to each other.(a) If
A large ground-based transmitter radiates 10 kW and communicates with a mobile receiving station that dissipates 1mW on the matched load of its antenna. The receiver (not having moved) now transmits
Consider an n-element broadside linear array. Increasing the number of elements has the effect of narrowing the main beam. Demonstrate this by evaluating the separation in ϕ between the zeros on
In a linear endfire array of n elements, a choice of current phasing that improves the directivity is given by the Hansen€“Woodyard condition:where the plus or minus sign choices give
A six-element linear dipole array has element spacing d = λ/2.(a) Select the appropriate current phasing, ξ, to achieve maximum radiation along ϕ = ±60◦.(b) With the phase set as in part (a),
Consider a linear endfire array, designed for maximum radiation intensity at ϕ = 0, using ξ and d values as suggested in Example 14.5. Determine an expression for the front-to-back ratio (defined
A turnstile antenna consists of two crossed dipole antennas, positioned in this case in the xy plane. The dipoles are identical, lie along the x and y axes, and are both fed at the origin. Assume
In the two-element endfire array of Example 14.4, consider the effect of varying the operating frequency, f, away from the original design frequency, f0, while maintaining the original current
Design a two-element dipole array that will radiate equal intensities in the ϕ = 0, π/2, π, and 3π/2 directions in the H plane. Specify the smallest relative current phasing, ξ, and the smallest
Consider a lossless half-wave dipole in free space, with radiation resistance, Rrad = 73 ohms, and maximum directivity Dmax = 1.64. If the antenna carries a 1-A current amplitude,(a) How much total
For a dipole antenna of overall length, 2ℓ = 1.5λ,(a) Evaluate the locations in θ at which the zeros and maxima in the E-plane pattern occur;(b) Determine the sidelobe level, as per the
For a dipole antenna of overall length 2„“ = 1.3Î», determine the locations in Î¸ and the peak intensity of the sidelobes, expressed as a fraction of the main lobe
For a dipole antenna of overall length 2ℓ = λ, evaluate the maximum directivity in decibels, and the half-power beamwidth.
The radiation field of a certain short vertical current element is Eθs = (20/r) sin θ e−j10πr V/m if it is located at the origin in free space.(a) Find Eθs at P(r = 100, θ = 90◦, ϕ =
Find the zeros in θ for the E-plane pattern of a dipole antenna for which(a) ℓ = λ;(b) 2ℓ = 1.3λ. Use Figure 14.8 as a guide.
A monopole antenna extends vertically over a perfectly conducting plane, and has a linear current distribution. If the length of the antenna is 0.01λ, what value of I0 is required to(a) Provide a
A dipole antenna in free space has a linear current distribution with zero current at each end, and with peak current I0 at the enter. If the length d is 0.02λ, what value of I0 is required to(a)
Show that the chord length in the E-plane plot of Figure 14.4 is equal to b sin Î¸, where b is the circle diameter. dA = r-dQ = r² sinOd@do
A short current element has d = 0.03λ. Calculate the radiation resistance that is obtained for each of the following current distributions:(a) Uniform, I0;(b) Linear, I (z) = I0(0.5d −
Consider the term in Eq. (14) (or in Eq. (10)) that gives the 1/r 2 dependence in the Hertzian dipole magnetic field. Assuming this term dominates and that kr << 1, show that the resulting
Write the Hertzian dipole electric field whose components are given in Eqs. (15) and (16) in the near zone in free space where kr << 1. In this case, only a single term in each of the two

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