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Chapter: 14 -Problem: 16 >> For a dipole antenna of overall length, 2? = 1.5?,(a) Evaluate the locations in ? at which the zeros and maxima in the E-plane pattern occur;(b) Determine the sidelobe level, as per the definition in Problem 14.14;(c) Determine the maximum directivity.
Answer Preview: a In this case k 2 1 5 2 1 5 The angular intensity distribution is given by the square of the patter…

, Chapter: 5 -Problem: 17 >> Given the potential field V = 100xz/(x2 + 4) V in free space:(a) Find D at the surface z = 0.(b) Show that the z = 0 surface is an equipotential surface.(c) Assume that the z = 0 surface is a conductor and find the total charge on that portion of the conductor defined by 0 < x < 2,?3 < y < 0.
Answer Preview: a Use At z 0 we use this to find D z 0 0 E z 0 100 0 x x 2 4 a …

, Chapter: 8 -Problem: 1 >> A point charge, Q = ?0.3?C 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, subject to the initial conditions at t = 0, v = 3×105ax m/s at the origin. At t = 3?s, find(a) The position P(x, y, z) of the charge;(b) The velocity v;(c) The kinetic energy of the charge.
Answer Preview: a The force on the charge is given by F qE and Newtons second law becomes descr…

, Chapter: 14 -Problem: 22 >> Revisit Problem 14.21, but with the current phase allowed to vary with frequency (this will automatically occur if the phase difference is established by a simple time delay between the feed currents). Now, the current phase difference will be ?' = ? f/ f0, where f0 is the original (design) frequency. Under this condition, radiation will maximize in the ? = 0 direction regardless of frequency (sho
Answer Preview: With the current phase and wavenumber both changing with frequency we …

, Chapter: 4 -Problem: 21 >> Let V = 2xy2z3 + 3 ln(x2 + 2y2 + 3z2) V in free space. Evaluate each of the following quantities at P(3, 2,?1)(a) V;(b) |V|;(c) E;(d) |E|;(e) aN;(f) D.
Answer Preview: a Substitute P directly to obtain V 15 0V b This will be just 1…

, Chapter: 7 -Problem: 1 >> (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 if both filaments are present.
Answer Preview: a Applying the Biot Savart Law we obtain Using integral tables th…

, Chapter: 4 -Problem: 15 >> Two uniform line charges, 8 nC/m each, are located at x = 1, z = 2, and at x = ?1, y = 2 in free space. If the potential at the origin is 100 V, find V at P(4, 1, 3).
Answer Preview: Two uniform line charges 8 nC m each are located at x 1 z 2 and at x 1 y …

, Chapter: 7 -Problem: 23 >> Given the field H = 20?2a? A/m:(a) Determine the current density J.(b) Integrate J over the circular surface ? ? 1, 0 < ? < 2?, z = 0, to determine the total current passing through that surface in the az direction.(c) Find the total current once more, this time by a line integral around the circular path ? = 1, 0 < ? < 2?, z = 0.
Answer Preview: a This is found through the curl of H which simplifies t…

, Chapter: 7 -Problem: 3 >> Two semi-infinite filaments on the z axis lie in the regions ? ? < z < ?a and a < z < ?. Each carries a current I in the az direction.(a) Calculate H as a function of ? and ? at z = 0.(b) What value of a will cause the magnitude of H at ? = 1, z = 0, to be one-half the value obtained for an infinite filament?
Answer Preview: a One way to do this is to use the field from an infinite l…

, Chapter: 14 -Problem: 25 >> A six-element linear dipole array has element spacing d = ?/2.(a) Select the appropriate current phasing, ?, to achieve maximum radiation along ? = ±60?.(b) With the phase set as in part (a), evaluate the intensities (relative to the maximum) in the broadside and endfire directions.
Answer Preview: a With d 2 we have kd and in the H plane kd cos cos We set t…

, Chapter: 5 -Problem: 10 >> A large brass washer has a 2-cm inside diameter, a 5-cm outside diameter, and is 0.5 cm thick. Its conductivity is ? = 1.5 × 107 S/m. The washer is cut in half along a diameter, and a voltage is applied between the two rectangular faces of one part. The resultant electric field in the interior of the half-washer is E = (0.5/?) a? V/m in cylindrical coordinates, where the z axis is the axis of the
Answer Preview: a First we orient the washer in the x y plane with the cut faces aligned wi…

, Chapter: 5 -Problem: 22 >> The line segment x = 0,?1 ? y ? 1, z = 1, carries a linear charge density ?L = ?|y|? C/m. Let z = 0 be a conducting plane and determine the surface charge density at:(a) (0, 0, 0);(b) (0, 1, 0).
Answer Preview: We consider the line charge to be made up of a string of different…

, Chapter: 14 -Problem: 18 >> Repeat Problem 14.17, but with a full-wave antenna (2? = ?). Numerical integrals may be necessary.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 power (in watts) is radiated?(b) How much power is intercepted by a 1-m2 aperture situated at distance r
Answer Preview: In this case k and the radiation resistance is found by applying Eq 65 where the answer was …

, Chapter: 4 -Problem: 7 >> Let G = 3xy2ax + 2zay Given an initial point P(2, 1, 1) and a final point Q(4, 3, 1), find ?G· dL using the path(a) Straight line: y = x ? 1, z = 1;(b) Parabola: 6y = x2 + 2, z = 1.
Answer Preview: a We obtain b …

, Chapter: 6 -Problem: 41 >> 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 V(a) Find the location of the 20 V equipotential surface.(b) Find Er,max.(c) Find
r if the surface charge density on the inner sphere is 1.0?C/m2.
Answer Preview: a Solving Laplaces equation gives us where V 0 100 a 5 and b 20 Setti…

, Chapter: 5 -Problem: 30 >> 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 + N2. The presence of an electric field E induces molecular dipole moments p1 and p2 within the individual species, whether mixed or not. Show that the dielectric constant of the composite material is gi
Answer Preview: We may write the total polarization vector as In terms of the suscept…

, Chapter: 5 -Problem: 11 >> Two perfectly conducting cylindrical surfaces of length  are located at ? = 3 and ? = 5 cm. The total current passing radially outward through the medium between the cylinders is 3 A dc.(a) Find the voltage and resistance between the cylinders, and E in the region between the cylinders, if a conducting material having ? = 0.05 S/m is present for 3 < ? < 5 cm.(b) Show that integrating the power dis
Answer Preview: a Given the current and knowing that it is radially directed we find the current densi…

, Chapter: 6 -Problem: 32 >> A uniform volume charge has constant density ?? = ?0 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 potential. Find(a) The potential everywhere;(b) The electric field intensity, E, everywhere.
Answer Preview: a Inside the sphere we solve Poissons equation assuming radial variation only We require that V …

, Chapter: 6 -Problem: 12 >> (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 thickness d and dielectric contant r. If ?r = 3, find d in terms of a such that the capacitance is twice that of part (a).
Answer Preview: a If we assume charge Q on the sphere the electric field will be The potential on …

, Chapter: 7 -Problem: 5 >> 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 Transcribed Image Text: (0, -1, 0) (0, 1, 0) 1A/ 1A
Answer Preview: We need an expression for H in cartesian coordinates We can star…

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, Chapter: 7 -Problem: 19 >> In spherical coordinates, the surface of a solid conducting cone is described by ? = ?/4 and a conducting plane by ? = ?/2. Each carries a total current I. The current flows as a surface current radially inward on the plane to the vertex of the cone, and then flows radially outward throughout the cross section of the conical conductor.(a) Express the surface current density as a function of r;(b)
Answer Preview: a This will be the total current divided by the circumfere…

, Chapter: 5 -Problem: 4 >> If volume charge density is given as ?v = (cos ?t)/r2 C/m2 in spherical coordinates, find J. It is reasonable to assume that J is not a function of ? or ?.
Answer Preview: We use the continuity equation 5 along with the assumption of no …

, Chapter: 7 -Problem: 32 >> The free space region defined by 1 < z < 4 cm and 2 < ? < 3 cm is a toroid of rectangular cross section. Let the surface at ? = 3 cm carry a surface current K = 2az kA/m.(a) Specify the current densities on the surfaces at ? = 2 cm, z = 1 cm, and z = 4 cm.(b) Find H everywhere.(c) Calculate the total flux within the toroid.
Answer Preview: a All surfaces must carry equal currents With this …

, Chapter: 5 -Problem: 26 >> A semiconductor sample has a rectangular cross section 1.5 by 2.0 mm, and a length of 11.0 mm. The material has electron and hole densities of 1.8 × 1018 and 3.0 × 1015 m?3, respectively. If ?e = 0.082 m2/V · s and ?h = 0.0021 m2/ V· s, find the resistance offered between the end faces of the sample.
Answer Preview: Using the given values along with th…

, Chapter: 6 -Problem: 6 >> 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 connected to a battery that applies a voltage V0 between plates. With the battery left connected, the plates are moved apart to a distance of 10d. Determine by what factor each of the following quantities cha
Answer Preview: The ordering of parameters is changed over that in Problem 6 4 as the p…

, Chapter: 12 -Problem: 16 >> In Figure 12.5, let regions 2 and 3 both be of quarter-wave thickness. Region 4 is glass, having refractive index, n4= 1.45; region 1 is air. (a) Find ηin,b.(b) Find ηin,a.(c) Specify a relation between the four intrinsic impedances that will enable total transmission of waves incident from the left into region 4.(d) Specify refractive index values for regions 2 and 3 that will accomplish the co
Answer Preview: a Since region 3 is a quarterwave layer 3 l b 2 and 47 reduces to inb 2 3 4 b Again with regio…

, Chapter: 7 -Problem: 39 >> 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 Vm if Vm = 0 at P(0.1, 0.2, 0.3);(c) Find B;(d) Obtain an expression for A if A = 0 at P.
Answer Preview: a Since we have parallel current sheets carrying eq…

, Chapter: 5 -Problem: 32 >> Two equal but opposite-sign point charges of 3 ?C are held x meters apart by a spring that provides a repulsive force given by Fsp = 12(0.5 ? x) N. Without any force of attraction, the spring would be fully extended to 0.5 m.(a) Determine the charge separation.(b) What is the dipole moment?
Answer Preview: a The Coulomb and spring forces must be equal in magnitude …

, Chapter: 6 -Problem: 38 >> Repeat Problem 6.37, but with the dielectric only partially filling the volume, within 0 < ? < ?, and with free space in the remaining volume. In ProblemCoaxial conducting cylinders are located at ? = 0.5 cm and ? = 1.2 cm. The region between the cylinders is filled with a homogeneous perfect dielectric. If the inner cylinder is at 100 V and the outer at 0 V, find
Answer Preview: We note that the dielectric changes with and not with Also since E is radiall…

, Chapter: 11 -Problem: 12 >> Describe how the attenuation coefficient of a liquid medium, assumed to be a good conductor, could be determined through measurement of wavelength in the liquid at a known frequency. What restrictions apply? Could this method be used to find the conductivity as well?
Answer Preview: In a good conductor we may use the approximation Ther…

, Chapter: 4 -Problem: 10 >> A sphere of radius a carries a surface charge density of ?s0 C/m2.(a) Find the absolute potential at the sphere surface.(b) A grounded conducting shell of radius b where b > a is now positioned around the charged sphere. What is the potential at the inner sphere surface in this case?
Answer Preview: a The setup for this is where from …

, Chapter: 7 -Problem: 18 >> A wire of 3 mm radius is made up of an inner material (0 < ? < 2 mm) for which ? = 107 S/m, and an outer material (2 mm < ? < 3 mm) for which ? = 4×107 S/m. If the wire carries a total current of 100 mA dc, determine H everywhere as a function of ?.
Answer Preview: Since the materials have different conductivities the current densities within them will differ …

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, Chapter: 5 -Problem: 25 >> Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are ?h = ??e = 6200T1.5e?7000/T C/m3. The functional dependence of the mobilities on temperature is given by ?h = 2.3 × 105T?2.7 m2/V · s and ?e = 2.1 × 105T?2.5 m2/V · s, where the temperature, T, is in degrees Kelvin. Find ? at:(a) 0?C;(b) 40?C;(c) 80?C.
Answer Preview: The conductivity will thus be Find at a With T 273K the …

, Chapter: 7 -Problem: 27 >> 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 = 4, 1 ? x ? 2, 3 ? z ? 5, in the az direction.(d) Show that the same result is obtained using the other side of Stokes’ theorem.
Answer Preview: a For this field the general curl expression in rectangular coordinates simplifies t…

, Chapter: 9 -Problem: 19 >> In Section 9.1, Faraday’s law was used to show that the field E = ?1/2 kB0ekt?a? results from the changing magnetic field B = B0ektaz.(a) Show that these fields do not satisfy Maxwell’s other curl equation.(b) If we let B0 = 1 T and k = 106 s?1, we are establishing a fairly large magnetic flux density in 1?s. Use the ? × H equation to show that the rate at which Bz should (but does not) change wit
Answer Preview: a B as stated is constant with position and so will have zero curl The elec…

, Chapter: 4 -Problem: 22 >> A line charge of infinite length lies along the z axis and carries a uniform linear charge density of ρ„“C/m. A perfectly conducting cylindrical shell, whose axis is the z axis, surrounds the line charge. The cylinder (of radius b), is at ground potential. Under these conditions, the potential function inside the cylinder (ρ < b) is given by where k is a constant.(a) Find k in terms of given or
Answer Preview: a At radius b b Find the electric field strength E …

, Chapter: 6 -Problem: 44 >> A potential field in free space is given as V = 100 ln tan(?/2) + 50 V.(a) Find the maximum value of |E?| on the surface ? = 40? for 0.1 < r < 0.8 m, 60? < ? < 90?.(b) Describe the surface V = 80 V.
Answer Preview: a First This will maximize at the smallest …

, Chapter: 6 -Problem: 14 >> 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.
Answer Preview: We use The above expression evaluates the capacitance of …

, Chapter: 6 -Problem: 37 >> Coaxial conducting cylinders are located at ? = 0.5 cm and ? = 1.2 cm. The region between the cylinders is filled with a homogeneous perfect dielectric. If the inner cylinder is at 100 V and the outer at 0 V, find(a) The location of the 20 V equipotential surface;(b) E? max;(c)
r if the charge per meter length on the inner cylinder is 20 nC/m.
Answer Preview: a From Eq 35 we have We seek at which V 20 V …

, Chapter: 14 -Problem: 29 >> 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 the transmitting antenna radiates 100 watts, how much power is dissipated by a matched load at the receiving antenna?(b) Suppose the receiving antenna is rotated by 45? while the two antennas remain in
Answer Preview: a To find the dissipated power use Eq 106 Since the antennas face each othe…

, Chapter: 10 -Problem: 13 >> The incident voltage wave on a certain lossless transmission line for which Z0 = 50? and ?p = 2 × 108 m/s is V+(z, t) = 200 cos(?t ? ?z) V.(a) Find ?.(b) Find I+(z, t). The section of line for which z > 0 is replaced by a load ZL = 50 + j30  at z = 0. Find:(c) L;(d) Vs? (z);(e) Vs at z = ?2.2 m.
Answer Preview: a We know v p so 2 10 8 628 10 8 rads b Since Z 0 is real we may wri…

, Chapter: 5 -Problem: 18 >> Two parallel circular plates of radius a are located at z = 0 and z = d. The top plate (z = d) is raised to potential V0; the bottom plate is grounded. Between the plates is a conducting material having radial-dependent conductivity, ?(?) = ?0?, where ?0 is a constant.(a) Find the ?-independent electric field strength, E, between plates.(b) Find the current density, J between plates.(c) Find the t
Answer Preview: a The integral of E between plates must give V 0 ind…

, Chapter: 7 -Problem: 31 >> The cylindrical shell defined by 1 cm < ? < 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 plane ? = 0, 0 < z < 1:(a) 0 < ? < 1.2 cm;(b) 1.0 cm < ? < 1.4 cm;(c) 1.4 cm < ? < 20 cm.
Answer Preview: a We first need to find J H and B The current density will be Next we find H at radius b…

, Chapter: 4 -Problem: 9 >> A uniform surface charge density of 20 nC/m2 is present on the spherical surface r = 0.6 cm in free space.(a) Find the absolute potential at P(r = 1 cm, ? = 25?, ? = 50?).(b) Find VAB, given points A(r = 2 cm, ? = 30?, ? = 60?) and B(r = 3 cm, ? = 45?, ? = 90?).
Answer Preview: a Since the charge density is uniform and is spherically symmetric the angul…

, Chapter: 6 -Problem: 15 >> 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 on the cylinder at a point nearest the plane.(b) Plane at a point nearest the cylinder;(c) Find the capacitance per unit length.
Answer Preview: Find the surface charge density on the a The cylinder will image across the plane producing an e…

, Chapter: 7 -Problem: 7 >> 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 plane.(a) Set up the Biot-Savart law and find H everywhere on the z axis;(b) Repeat part (a), but with the copper sheet occupying the entire x y plane.
Answer Preview: a First the contribution to the field at z from the current on the negative z axis …

, Chapter: 6 -Problem: 33 >> The functions V1(?, ?, z) and V2(?, ?, z) both satisfy Laplace’s equation in the region a < ? < b, 0 ? v < 2?, ?L < z < L; each is zero on the surfaces ? = b for ?L < z < L; z = ?L for a < ? < b; and z = L for a < ? < b; and each is 100 V on the surface ? = a for ?L < z < L.(a) In the region specified, is Laplace’s equation satisfied by the functions V1 + V2, V1 ? V2, V1 + 3, and V1V2?(b) On the b
Answer Preview: The functions V 1 z and V 2 z both satisfy Laplaces equa…

, Chapter: 7 -Problem: 43 >> 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 the proper zero reference and sketch the results on the figure.Figure 7.20 Transcribed Image Text: Ho
Answer Preview: We do this by first finding B within the outer conductor and then uncurling the …

, Chapter: 5 -Problem: 16 >> A coaxial transmission line has inner and outer conductor radii a and b. Between conductors (a < ? < b) lies a conductive medium whose conductivity is ?(?) = ?0/?, where ?0 is a constant. The inner conductor is charged to potential V0, and the outer conductor is grounded.(a) Assuming dc radial current I per unit length in z, determine the radial current density field J in A/m2.(b) Determine the el
Answer Preview: a This will be the current divided by the cross sectional area that is normal …

, Chapter: 7 -Problem: 21 >> 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 = b? az A/m2 where b is a constant.(a) What total current flows in the wire?(b) Find Hin (0 < ? < a), as a function of ?;(c) find Hout (? > a), as a function of ?;(d) verify your results of parts (b) and (c) by using? × H = J.
Answer Preview: a We integrate the current density over the wire cross se…

, Chapter: 7 -Problem: 2 >> A filamentary conductor is formed into an equilateral triangle with sides of length ? carrying current I. Find the magnetic field intensity at the center of the triangle.
Answer Preview: The magnetic field induction at OO …

, Chapter: 7 -Problem: 12 >> 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 7.22 (a) ˆ’0.2;(b) 0.2;(c) 0.4;(d) 0.75;(e) 1.2 m. Transcribed Image Text: Air 1.0 10 A/m? 0.7 Air 0.3 - 10 A/m? Ai
Answer Preview: The problem asks you to find H at various positions Before continuing we need to know how to find H for this type of current configuration The sketch …

, Chapter: 12 -Problem: 15 >> Sulfur hexafluoride (SF6) is a high-density gas that has refractive index, ns = 1.8 at a specified pressure, temperature, and wavelength. Consider the retro-reflecting prism shown in Fig. 12.16, that is immersed in SF6. Light enters through a quarter-wave antireflective coating and then totally reflects from the back surfaces of the glass. In principle, the beam should experience zero loss at the
Answer Preview: a We set the critical angle of total reflection equal to 45 which give…

, Capacitors tend to be more expensive as their capacitance and maximum voltage Vmax increase. The voltage Vmax is limited by the field strength at which the dielectric breaks down, EBD. Which of these dielectrics will give the largest CVmax product for equal plate areas?(a) Air: ?r = 1, EBD = 3 MV/m.(b) Barium titanate: ?r = 1200, EBD = 3 MV/m.(c) Silicon dioxide: ?r = 3.78, EBD = 16 MV/m.(d) Polye
Answer Preview: Solution :- Given data is, For an capacitor Capacitance ( C ) = ( E …

, Chapter: 5 -Problem: 20 >> Two point charges of ?100??C are located at (2, ?1, 0) and (2, 1, 0). The surface x = 0 is a conducting plane.(a) Determine the surface charge density at the origin.(b) Determine ?S at P(0, h, 0).
Answer Preview: a I will solve the general case first in which we fi…

, Chapter: 11 -Problem: 8 >> An electric field in free space is given in spherical coordinates as Es (r) = E0(r)e?jkr a? V/m.(a) Find Hs (r) assuming uniform plane wave behavior.(b) Find < S >.(c) Express the average outward power in watts through a closed spherical shell of radius r, centered at the origin.(d) Establish the required functional form of E0(r) that will enable the power flow in part c to be independent of radiu
Answer Preview: a Knowing that the cross product of E s with the complex conjugate of the pha…

, Chapter: 7 -Problem: 33 >> Use an expansion in rectangular coordinates to show that the curl of the gradient of any scalar field G is identically equal to zero.
Answer Preview: We begi…

, Chapter: 7 -Problem: 36 >> 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.
Answer Preview: a Show that A 0 b First A P 6a x 12a y Then using the …

, Chapter: 14 -Problem: 14 >> For a dipole antenna of overall length 2? = ?, evaluate the maximum directivity in decibels, and the half-power beamwidth.
Answer Preview: D max is found using Eq 64 with k and involves a numerical …

, Chapter: 12 -Problem: 23 >> Suppose that Ï• in Figure 12.17 is Brewster€™s angle, and that θ1is the critical angle. Find n0in terms of n1and n2. Transcribed Image Text: ?z ?? ?y ?2
Answer Preview: With the incoming ray at Brewsters angle the refracted angle of this ray measured f…

, Chapter: 14 -Problem: 27 >> 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 ? between the zeros on either side of the principal maximum at ? = 90?. Show that for large n this separation is approximated by ?? = 2?/L, where L = nd is the overall length of the array.
Answer Preview: For broadside operation 0 and with kd 2d we have With this condition …

, Chapter: 8 -Problem: 40 >> A coaxial cable has conductor radii a and b, where a < b. Material of permeability ?r ? 1 exists in the region a < ? < c, whereas the region c < ? < b is air filled. Find an expression for the inductance per unit length.
Answer Preview: In both regions the magnetic field will be H I2 a Am So the flux per unit l…

, Chapter: 7 -Problem: 17 >> 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 ? = 1 cm and ? = 0.5 cm, respectively. Calculate H at:(a) ? = 0.5 cm;(b) ? = 1.5 cm;(c) ? = 4 cm.(d) What current sheet should be located at ? = 4 cm so that H = 0 for all ? > 4 cm?
Answer Preview: a Here we are either just inside or just outside the first current sheet so both we wil…

, Chapter: 6 -Problem: 45 >> In free space, let ?? = 200
0/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 Gauss’s law and a line integral.
Answer Preview: a With r variation only we have or Integrate once or Our first bound…

, Chapter: 7 -Problem: 26 >> 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 ?S1 (? × H) · dS if H = 3? a? in cylindrical coordinates.
Answer Preview: First the intersection of the sphere with the x y plane …

, Chapter: 14 -Problem: 28 >> 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 back to the ground station. If the mobile unit radiates 100 W, what power is received (at a matched load) by the ground station?
Answer Preview: We can use Eq 93 and the fa…

, Chapter: 14 -Problem: 20 >> A two-element dipole array is configured to provide zero radiation in the broadside (? = ± 90?) and end-fire (? = 0, 180?) directions, but with maxima occurring at angles in between. Consider such a set-up with ? = ? at ? = 0 and ? = ?3? at ? = ?, with both values determined in the H-plane.(a) Verify that these values give zero broadside and end-fire radiation.(b) Determine the required relative c
Answer Preview: a For two elements the array function is given by Eq 81 with n 2 Thi…

, Chapter: 7 -Problem: 13 >> A hollow cylindrical shell of radius a is centered on the z axis and carries a uniform surface current density of Kaa?.(a) Show that H is not a function of ? or z.(b) Show that H? and H? are everywhere zero.(c) Show that Hz = 0 for ? > a.(d) Show that Hz = Ka for ? < a.(e) A second shell, ? = b, carries a current Kba?. Find H everywhere.
Answer Preview: a Consider this situation as illustrated in Fig 8 11 There sec 8 2 it was stated that the field will be entirely z directed We can see this by applyin…

, Chapter: 10 -Problem: 32 >> In Figure 10.17, let ZL= 250, Z0= 50, find the shortest attachment distance d and the shortest length d1of a short-circuited stub line that will provide a perfect match on the main line to the left of the stub. Express all answers in wavelengths. Transcribed Image Text: Z.
Answer Preview: The first step is to mark the normalized load admittance on the chart This will be y L 1z L 50250 …

, Chapter: 4 -Problem: 17 >> Uniform surface charge densities of 6 and 2 nC/m2 are present at ? = 2 and 6 cm, respectively, in free space. Assume V = 0 at ? = 4 cm, and calculate V at(a) ? = 5 cm;(b) ? = 7 cm.
Answer Preview: a Since V 0 at 4 cm the potential at 5 cm wi…

, Chapter: 5 -Problem: 27 >> 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 effective length of 7.1 × 10?19 m.(a) Find P.(b) Find ?r.
Answer Preview: a With all identical dipoles we have P …

, Chapter: 5 -Problem: 23 >> A dipole with p = 0.1az ?C · m is located at A(1, 0, 0) in free space, and the x = 0 plane is perfectly conducting.(a) Find V at P(2, 0, 1).(b) Find the equation of the 200 V equipotential surface in rectangular coordinates.
Answer Preview: a We use the far field potential for a z directed dipole The dipo…

, Chapter: 6 -Problem: 20 >> 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 and construct a curvilinear-square map for its interior.(b) Assume ? = ?0 and estimate C per meter length.
Answer Preview: a The result below could still be improved a little but is ne…

, Chapter: 4 -Problem: 24 >> A certain spherically symmetric charge configuration in free space produces an electric field given in spherical coordinates by where ρ0 is a constant.(a) Find the charge density as a function of position.(b) Find the absolute potential as a function of position in the two regions, r ‰¤ 10 and r ‰¥ 10.(c) Check your result of part b by using the gradient.(d) Find the stored energy in the charge b
Answer Preview: a Find the charge density as a function of position b Find the absolute potential as a …

, Chapter: 6 -Problem: 18 >> 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 for the actual sketch if symmetry is considered. As a check, compute the capacitance per meter both from your sketch and from the exact formula. Assume ?r = 1.
Answer Preview: Symmetry allows us to plot the field lines and equipotentials over just the first quadrant as …

, Chapter: 7 -Problem: 35 >> A current sheet, K = 20 az A/m, is located at ? = 2, and a second sheet, K = ?10az A/m, is located at ? = 4.(a) Let Vm = 0 at P(? = 3, ? = 0, z = 5) and place a barrier at ? = ?. Find Vm(?, ?, z) for ?? < ? Answer Preview: a Since the current is cylindrically symmetric we know that H I 2 a wh…

, Chapter: 6 -Problem: 22 >> Two conducting plates, each 3 × 6 cm, and three slabs of dielectric, each 1 × 3 × 6 cm, and having dielectric constants of 1, 2, and 3, are assembled into a capacitor with d = 3 cm. Determine the two values of capacitance obtained by the two possible methods of assembling the capacitor.
Answer Preview: The two possible configurations are 1 all slabs positi…

, Chapter: 5 -Problem: 13 >> A hollow cylindrical tube with a rectangular cross section has external dimensions of 0.5 in. by 1 in. and a wall thickness of 0.05 in. Assume that the material is brass, for which ? = 1.5 × 107 S/m. A current of 200 A dc is flowing down the tube.(a) What voltage drop is present across a 1 m length of the tube?(b) Find the voltage drop if the interior of the tube is filled with a conducting materi
Answer Preview: a Converting all measurements to meters the tube resistance over a 1 m length …

, Chapter: 7 -Problem: 8 >> 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 Transcribed Image Text: (sin a2 – sina1)as 4?? ? Point 2
Answer Preview: The Biot Savart law reads The integra…

, Chapter: 6 -Problem: 46 >> By appropriate solution of Laplace’s and Poisson’s equations, determine the absolute potential at the center of a sphere of radius a, containing uniform volume charge of density ?0. Assume permittivity ?0 everywhere.
Answer Preview: With radial dependence only Poissons equation applicable to …

, Chapter: 4 -Problem: 3 >> If E = 120a? V/m, find the incremental amount of work done in moving a 50-?C charge a distance of 2 mm from(a) P(1, 2, 3) toward Q(2, 1, 4);(b) Q(2, 1, 4) toward P(1, 2, 3).
Answer Preview: If E 120 a V m find the incremental amount of work done in moving a 50 m charge a distance of 2 mm f…

, Chapter: 3 -Problem: 11 >> In cylindrical coordinates, let ?? = 0 for? < 1 mm, ?? = 2 sin(2000 ??) nC/m3 for 1 mm < ? < 1.5 mm, and ?? = 0 for ? > 1.5 mm. Find D everywhere.
Answer Preview: Since the charge varies only with radius and is in the form of a cylinder sy…

, Chapter: 5 -Problem: 2 >> Given J = ?10?4(yax + xay)A/m2, find the current crossing the y = 0 plane in the ?ay direction between z = 0 and 1, and x = 0 and 2.
Answer Preview: At y 0 J x 0 10 4 xa y …

, Chapter: 6 -Problem: 34 >> 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 plates are at ground potential. By solving Laplace’s and Poisson’s equations, find(a) V(z) for 0 < z < d;(b) the electric field intensity for 0 < z < d.
Answer Preview: a In Region 1 z b we solve Poissons equation assuming z variation only In Region 2 z b …

, Chapter: 4 -Problem: 13 >> Three identical point charges of 4 pC each are located at the corners of an equilateral triangle 0.5 mm on a side in free space. How much work must be done to move one charge to a point equidistant from the other two and on the line joining them?
Answer Preview: This will be the magnitude of th…

, Chapter: 6 -Problem: 40 >> 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 axis. Potential V0 is on the top plate; the bottom plate is grounded. Dielectric having radially dependent permittivity fills the region between plates. The permittivity is given by (?) = 0(1 + ?2/a2). F
Answer Preview: a Since varies in the direction normal to E Laplaces equation applies and we write With the …

, Chapter: 4 -Problem: 23 >> It is known that the potential is given as V = 80?0.6 V. Assuming free space conditions, find.(a) E;(b) the volume charge density at ? = 0.5 m;(c) The total charge lying within the closed surface ? = 0.6, 0 < z < 1.
Answer Preview: It is known that the potential is given as V 80 6 V Assuming free space condi…

, Chapter: 14 -Problem: 23 >> 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 that equal currents are supplied to each antenna and that a zero phase reference is applied to the x-directed antenna. Determine the relative phase, ?, of the y-directed antenna so that the net radiated e
Answer Preview: When looking at the field along the z axis the expressio…

, Chapter: 6 -Problem: 28 >> Show that in a homogeneous medium of conductivity ?, the potential field V satisfies Laplace’s equation if any volume charge density present does not vary with time.
Answer Preview: We begin with the continuity equation Eq 5 Ch…

, Chapter: 14 -Problem: 19 >> Design a two-element dipole array that will radiate equal intensities in the ? = 0, ?/2, ?, and 3?/2 directions in the H plane. Specify the smallest relative current phasing, ?, and the smallest element spacing, d.
Answer Preview: The array function is given by Eq 81 with n 2 This has periodic maxima occurring …

, Chapter: 6 -Problem: 21 >> 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 good-sized drawing of this transmission line, say with a = 2.5 cm, and then prepare a curvilinear-square plot of the electrostatic field between the conductors. (b) Use the map to calculate the capacitance pe
Answer Preview: a Some improvement is possible depending on ho…

, Chapter: 5 -Problem: 6 >> In spherical coordinates, a current density J = ?k/(r sin ?) a? A/m2 exists in a conducting medium, where k is a constant. Determine the total current in the az direction that crosses a circular disk of radius R, centered on the z axis and located at(a) z = 0;(b) z = h.
Answer Preview: a z 0 b z h Integration over a disk means that we use …

, Chapter: 6 -Problem: 10 >> 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 radius of 2.7 mm, and with a thickness of 3.0 mm. The spacers are located every 2 cm down the cable.(a) By what factor do the spacers increase the capacitance per unit length?(b) If 100 V is maintained
Answer Preview: a The net capacitance can be constructed as a composite quantity composed of weighted contributions …

, Chapter: 5 -Problem: 14 >> A rectangular conducting plate lies in the xy plane, occupying the region 0 < x < a, 0 < y < b. An identical conducting plate is positioned directly above and parallel to the first, at z = d. The region between plates is filled with material having conductivity ?(x) = ?0e?x/a, where ?0 is a constant. Voltage V0 is applied to the plate at z = d; the plate at z = 0 is at zero potential. Find, in ter
Answer Preview: a We know that E will be z directed but the conductivity v…

, Chapter: 14 -Problem: 13 >> The radiation field of a certain short vertical current element is E?s = (20/r) sin ? e?j10?r V/m if it is located at the origin in free space.(a) Find E?s at P(r = 100, ? = 90?, ? = 30?).(b) Find E?s at P(100, 90?, 30?) if the vertical element is located at A(0.1, 90?, 90?).(c) Find E?s at P(100, 90?, 30?) if identical vertical elements are located at A(0.1, 90?, 90?) and B(0.1, 90?, 270?).
Answer Preview: a Substituting these values into the given formula find b This places the element on the y axis at y …

, Chapter: 7 -Problem: 20 >> 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 two ends. Within the conductor, H = 105?2a? A/m.(a) Find ? as a function of ?.(b) What is the resistance between the two ends?
Answer Preview: a Start by finding J from H by taking the curl With H di…

, Chapter: 14 -Problem: 12 >> Find the zeros in ? for the E-plane pattern of a dipole antenna for which(a) ? = ?;(b) 2? = 1.3?. Use Figure 14.8 as a guide.
Answer Preview: a We look for zeros in the pattern function Eq 59 for which k 2 2 Zeros will occur whenever …

, Chapter: 7 -Problem: 22 >> A solid cylinder of radius a and length L, where L >>a, contains volume charge of uniform density ?0 C/m3. The cylinder rotates about its axis (the z axis) at angular velocity ? rad/s.(a) Determine the current density J as a function of position within the rotating cylinder.(b) Determine H on-axis by applying the results of Problem 7.6.(c) Determine the magnetic field intensity H inside and outsid
Answer Preview: a Use J 0 v 0 a A m 2 b It helps initially to obtain the field on axis To do this we use the result …

, Chapter: 12 -Problem: 1 >> A uniform plane wave in air, E+x1 = E+x10 cos(1010t ? ?z) V/m, is normally incident on a copper surface at z = 0. What percentage of the incident power density is transmitted into the copper?
Answer Preview: We need to find the reflection coefficient The intrinsic impedance of copper …

, Chapter: 9 -Problem: 7 >> The rails in Figure 9.6 each have a resistance of 2.2/m. The bar moves to the right at a constant speed of 9 m/s in a uniform magnetic field of 0.8 T. Find I (t), 0 < t < 1 s, if the bar is at x = 2 m at t = 0 and(a) A 0.3 Ω resistor is present across the left end with the right end open-circuited;(b) A 0.3 Ω resistor is present across each end.
Answer Preview: a The flux in the left hand closed loop is l B area 0 8 0 2 2 9t Then emf l dl dt 0 16 9 1 44 V …

, Chapter: 10 -Problem: 27 >> The characteristic admittance (Y0 = 1/Z0) of a lossless transmission line is 20 mS. The line is terminated in a load YL = 40 ? j20 mS. Use the Smith chart to find(a) s;(b) Yin if l = 0.15?;(c) the distance in wavelengths from YL to the nearest voltage maximum.
Answer Preview: a We first find the normalized load admittance which is y L Y L Y 0 2 j1 This is plotted on the Smit…

, Chapter: 8 -Problem: 4 >> Show that a charged particle in a uniform magnetic field describes a circular orbit with an orbital period that is independent of the radius. Find the relationship between the angular velocity and magnetic flux density for an electron (the cyclotron frequency).
Answer Preview: A circular orbit can be established if the magnetic force on the particle is balanced by …

, Chapter: 11 -Problem: 7 >> The phasor magnetic field intensity for a 400 MHz uniform plane wave propagating in a certain lossless material is (2ay ? j5az)e?j25x A/m. Knowing that the maximum amplitude of E is 1500 V/m, find ?, ?, ?, ?p, ?r, ?r, and H(x, y, z, t).
Answer Preview: First from the phasor expression we identify 25 m 1 from the argument of the exponential func…

, Chapter: 10 -Problem: 10 >> Two lossless transmission lines having different characteristic impedances are to be joined end to end. The impedances are Z01 = 100 and Z03 = 25 ?. The operating frequency is 1 GHz.(a) Find the required characteristic impedance, Z02, of a quarter-wave section to be inserted between the two, which will impedance-match the joint, thus allowing total power transmission through the three lines.(b) Th
Answer Preview: a The required inpedance will be Z 02 Z 01 Z 03 100 25 50 …

, Chapter: 6 -Problem: 30 >> 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 ?0 C/m3 and has permittivity ?. Both plates are held at ground potential.(a) Determine the potential field between plates.(b) Determine the electric field intensity E between plates.(c) Repeat parts (a) and (b) for the case of the plat
Answer Preview: a We solve Poissons equation under the assumption that V varies only with z At z 0 V 0 and so C 2 …

, Chapter: 6 -Problem: 8 >> 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 axis. Potential V0 is on the top plate; the bottom plate is grounded. Dielectric having radially dependent permittivity fills the region between plates. The permittivity is given by ?(?) = ?0(1 + ?2/a2).
Answer Preview: a Since does not vary in the z direction and since we must always obtain V …

, Chapter: 11 -Problem: 2 >> A 10 GHz uniform plane wave propagates in a lossless medium for which ?r = 8 and ?r = 2. Find(a) ?p;(b) ?;(c) ?;(d) Es;(e) Hs;(f) S.
Answer Preview: a v p c r 3 10 8 5 1 34 10 8 m s b v p 2 10 8 1 34 10 8 4 69 m 1 c 2 …

, Chapter: 8 -Problem: 39 >> Conducting planes in air at z = 0 and z = d carry surface currents of ±K0ax A/m.(a) Find the energy stored in the magnetic field per unit length (0 < x < 1) in a width w(0 < y < w).(b) Calculate the inductance per unit length of this transmission line from WH = 1/2 LI2, where I is the total current in a width w in either conductor. (c) Calculate the total flux passing through the rectangle 0 < x <
Answer Preview: a First assuming current flows in the a x direction in the sheet at z d and in a x in the …

, Chapter: 8 -Problem: 6 >> Show that the differential work in moving a current element IdL through a distance dl in a magetic field B is the negative of that done in moving the element Idl through a distance dL in the same field.
Answer Preview: The two differential work quantities …

, Chapter: 1 -Problem: 1 >> Given the vectors M = ?10ax + 4ay ? 8az and N = 8ax + 7ay ? 2az, find:(a) A unit vector in the direction of ?M+ 2N;(b) The magnitude of 5ax + N ? 3M; (c) |M||2N|(M+ N).
Answer Preview: Given the vectors M 10a x 4a y 8a z and N 8a x 7a y 2a z find a A …

, Chapter: 3 -Problem: 26 >> If we have a perfect gas of mass density ?m kg/m3, and we assign a velocity U m/s to each differential element, then the mass flow rate is ?mU kg/(m2 ? s). Physical reasoning then leads to the continuity equation, ? · (?mU) = ???m/?t.(a) Explain in words the physical interpretation of this equation.(b) Show that ?s ?m U· dS = ?dM/dt, where M is the total mass of the gas within the constant closed
Answer Preview: a The quantity m U is the flow or flux density of mass T…

, Chapter: 11 -Problem: 1 >> Show that Exs= Aej(k0z+Ï•)is a solution of the vector Helmholtz equation, Eq. (30), for k0= ωˆšÎ¼0ˆˆ0and any Ï• and A. Transcribed Image Text: d²Exs = -k,Es. dz?
Answer Preview: We take …

, Chapter: 4 -Problem: 28 >> Use the electric field intensity of the dipole [Section 4.7, Eq. (35)] to find the difference in potential between points at θaand θb, each point having the same r and ϕ coordinates. Under what conditions does the answer agree with Eq. (33), for the potential at θa?Eq. (35)Eq. (33) Transcribed Image Text:
Answer Preview: We perform a line integral of Eq 36 along an arc of constant r and This result a…

, Chapter: 10 -Problem: 3 >> The characteristic impedance of a certain lossless transmission line is 72. If L = 0.5?H/m, find(a) C;(b) ?p;(c) ? if f = 80 MHz.(d) The line is terminated with a load of 60 ?. Find ? and s.
Answer Preview: a Use Z 0 L C or v p c if …

, Chapter: 3 -Problem: 16 >> An electric flux density is given by D = D0 a?, where D0 is a given constant.(a) What charge density generates this field?(b) For the specified field, what total charge is contained within a cylinder of radius a and height b, where the cylinder axis is the z axis?
Answer Preview: a Charge density is found by taking the divergen…

, Chapter: 10 -Problem: 28 >> The wavelength on a certain lossless line is 10 cm. If the normalized input impedance is zin = 1 + j 2, use the Smith chart to determine(a) s;(b) zL, if the length of the line is 12 cm;(c) xL, if zL = 2 + j xL where xL > 0.
Answer Preview: a We begin by marking z in on the chart see below and setting the compass at its distance from the o…

, Chapter: 8 -Problem: 2 >> Compare the magnitudes of the electric and magnetic forces on an electron that has attained a velocity of 107 m/s. Assume an electric field intensity of 105 V/m, and a magnetic flux density associated with that of the Earth’s magnetic field in temperate latitudes, 0.5 gauss.
Answer Preview: We use the Lorentz Law F F e F m …

, Chapter: 2 -Problem: 23 >> Given the surface charge density, ?s = 2?C/m2, existing in the region ? < 0.2 m, z = 0, find E at(a) PA(? = 0, z = 0.5);(b) PB(? = 0, z = ? 0.5). Show that(c) The field along the z axis reduces to that of an infinite sheet charge at small values of z;(d) The z axis field reduces to that of a point charge at large values of z.
Answer Preview: Given the surface charge density s 2 C m 2 in the region 0 2 m z 0 Find E at a PA 0 z 0 5 First we r…

, Chapter: 3 -Problem: 15 >> Volume charge density is located as follows: ?? = 0 for ? < 1 mm and for ? > 2 mm, ?? = 4? ?C/m3 for 1 < ? < 2 mm.(a) Calculate the total charge in the region 0 < ? < ?1, 0 < z < L, where 1 < ?1 < 2 mm.(b) Use Gauss’s law to determine D? at ? = ?1.(c) Evaluate D? at ? = 0.8 mm, 1.6 mm, and 2.4 mm.
Answer Preview: a We find b Gauss law states that 2 1 LD Q where Q is the result of part a So with …

, Chapter: 10 -Problem: 34 >> The lossless line shown in Figure 10.35 is operating with λ = 100 cm. If d1= 10 cm, d = 25 cm, and the line is matched to the left of the stub, what is ZL? Transcribed Image Text: d, s.c. Zg = 300 2 Z, = 300 2
Answer Preview: For the line to be matched it is required that the sum of the normalized input admitt…

, Chapter: 12 -Problem: 21 >> A right-circularly polarized plane wave in air is incident at Brewster’s angle onto a semi-infinite slab of plexiglas (?'r = 3.45, ?"r
= 0).(a) Determine the fractions of the incident power that are reflected and transmitted.(b) Describe the polarizations of the reflected and transmitted waves.
Answer Preview: a In plexiglas Brewsters angle is B 1 tan 1 r2 r1 tan 1 3 45 61 7 Then the angle of refraction is …

, Chapter: 1 -Problem: 24 >> Two unit vectors, a1 and a2, lie in the xy plane and pass through the origin. They make angles ?1 and ?2, respectively, with the x axis(a) Express each vector in rectangular components;(b) Take the dot product and verify the trigonometric identity, cos(?1 ? ?2) = cos ?1 cos ?2 + sin ?1 sin ?2;(c) Take the cross product and verify the trigonometric identity sin(?2 ? ?1) = sin ?2 cos ?1 ? cos ?2 sin
Answer Preview: Two unit vectors a 1 and a 2 lie in the x y plane and pass through the origin They …

, Chapter: 10 -Problem: 40 >> In the charged line of Figure 10.25, the characteristic impedance is Z0= 100 , and Rg= 300 Ω. The line is charged to initial voltage, V0= 160 V, and the switch is closed at t = 0. Determine and plot the voltage and current through the resistor for time 0 < t < 8l/ν (four round-trips). This problem accompanies Example 10.12 as the other special case of the basic charged-line problem, in which now
Answer Preview: On closing the switch the initial voltage wave is Now with g 1 2 a…

, Chapter: 1 -Problem: 14 >> Given that A + B + C = 0, where the three vectors represent line segments and extend from a common origin, must the three vectors be coplanar? If A + B + C + D = 0, are the four vectors coplanar?
Answer Preview: Given that A B C 0 where the three vectors represent line segments and extend …

, Chapter: 8 -Problem: 5 >> A rectangular loop of wire in free space joins point A(1, 0, 1) to point B(3, 0, 1) to point C(3, 0, 4) to point D(1, 0, 4) to point A. The wire carries a current of 6 mA, flowing in the az direction from B to C. A filamentary current of 15 A flows along the entire z axis in the az direction.(a) Find F on side BC.(b) Find F on side AB.(c) Find Ftotal on the loop.
Answer Preview: a Find F on side BC Thus b The field from the long wire now varies with position along th…

, Chapter: 11 -Problem: 6 >> A uniform plane wave has electric field Es = (Ey0 ay ? Ez0 az) e??x e?j?x V/m. The intrinsic impedance of the medium is given as ? = |?| ej?, where ? is a constant phase.(a) Describe the wave polarization and state the direction of propagation.(b) Find Hs.(c) Find E(x, t) and H(x, t).(d) Find < S > in W/m2.(e) Find the time-average power in watts that is intercepted by an antenna of rectangular cr
Answer Preview: a The wave is linearly polarized in the y z plane and propagates in the forward x directi…

, Chapter: 11 -Problem: 9 >> A certain lossless material has ?r = 4 and ?r = 9. A 10-MHz uniform plane wave is propagating in the ay direction with Ex0 = 400 V/m and Ey0 = Ez0 = 0 at P(0.6, 0.6, 0.6) at t = 60 ns. Find(a) ?, ?, ?p, and ?;(b) E(y, t);(c) H(y, t).
Answer Preview: a For a uniform plane wave Then 2 2 0 4 5m Next Finally b We are given t…

, Chapter: 11 -Problem: 4 >> Small antennas have low efficiencies (as will be seen in Chapter 14), and the efficiency increases with size up to the point at which a critical dimension of the antenna is an appreciable fraction of a wavelength, say ?/8.(a) An antenna that is 12 cm long is operated in air at 1 MHz. What fraction of a wavelength long is it?(b) The same antenna is embedded in a ferrite material for which ?r = 20 a
Answer Preview: a The free space wave…

, Chapter: 8 -Problem: 10 >> A planar transmission line consists of two conducting planes of width b separated d m in air, carrying equal and opposite currents of I A. If b >> d, find the force of repulsion per meter of length between the two conductors.
Answer Preview: Take the current in the top plate in the positive z directio…

, Chapter: 2 -Problem: 20 >> A line charge of uniform charge density ?0 C/m and of length ? is oriented along the z axis at ? ?/2 < z < ?/2.(a) Find the electric field strength, E, in magnitude and direction at any position along the x axis.(b) With the given line charge in position, find the force acting on an identical line charge that is oriented along the x axis at ?/2 < x < 3 ?/2.
Answer Preview: A line charge of uniform charge density 0 C m and of length is oriented along the z axis at 2 z 2 a …

, Chapter: 11 -Problem: 3 >> An H field in free space is given as H(x, t) = 10 cos(108t ? ?x)ay A/m. Find(a) ?;(b) ?;(c) E(x, t) at P(0.1, 0.2, 0.3) at t = 1 ns.
Answer Preview: a Since we have a uniform plane wave c where we identify 10 8 sec 1 Thus 10 8 3 10 8 0 33 …

, Chapter: 2 -Problem: 16 >> Within a region of free space, charge density is given as ?? = ?0 rcos ?/a C/m3, where ?0 and a are constants. Find the total charge lying within(a) The sphere, r ? a;(b) The cone, r ? a, 0 ? ? ? 0.1?;(c) The region, r ? a, 0 ? ? ? 0.1?, 0 ? ? ? 0.2?.
Answer Preview: Within a region of free space charge density is given …

, Chapter: 10 -Problem: 23 >> The normalized load on a lossless transmission line is 2 + j 1. Let ? = 20 m and make use of the Smith chart to find(a) The shortest distance from the load to a point at which zin = rin + j0, where rin > 0;(b) zin at this point.(c) The line is cut at this point and the portion containing zL is thrown away. A resistor r = rin of part (a) is connected across the line. What is s on the remainder of t
Answer Preview: a Referring to the figure below we start by marking the given z L on the chart and drawing a line fr…

, Chapter: 12 -Problem: 9 >> Region 1, z < 0, and region 2, z > 0, are both perfect dielectrics (? = ?0,


= 0). A uniform plane wave traveling in the az direction has a radian frequency of 3 × 1010 rad/s. Its wavelengths in the two regions are ?1 = 5 cm and ?2 = 3 cm. What percentage of the energy incident on the boundary is(a) Reflected;(b) Transmitted?(c) What is the standing wave ratio in region 1?
Answer Preview: a We first note that Therefore r1 r2 2 1 2 Then wit…

, Chapter: 4 -Problem: 31 >> A potential field in free space is expressed as V = 20/(xyz)V.(a) Find the total energy stored within the cube 1 < x, y, z < 2.(b) What value would be obtained by assuming a uniform energy density equal to the value at the center of the cube?
Answer Preview: a We integrate the energy density over the cube volume wh…

, Chapter: 12 -Problem: 22 >> A dielectric waveguide is shown in Figure 12.17 with refractive indices as labeled. Incident light enters the guide at angle Ï• from the front surface normal as shown. Once inside, the light totally reflects at the upper n1ˆ’ n2interface, where n1> n2. All subsequent reflections from the upper and lower boundaries will be total as well, and so the light is confined to the guide. Express, in terms
Answer Preview: From the illustration we see that 1 maximizes when 1 is at its minimum value This minimu…

, Chapter: 1 -Problem: 2 >> Vector A extends from the origin to (1, 2, 3), and vector B extends from the origin to (2, 3,?2). Find(a) the unit vector in the direction of (A ? B);(b) the unit vector in the direction of the line extending from the origin to the midpoint of the line joining the ends of A and B.
Answer Preview: Vector A extends from the origin to 1 2 3 and vector B fr…

, Chapter: 9 -Problem: 22 >> In a sourceless medium in which J = 0 and ?? = 0, assume a rectangular coordinate system in which E and H are functions only of z and t. The medium has permittivity ? and permeability ?.(a) If E = Exax and H = Hyay, begin with Maxwell’s equations and determine the second-order partial differential equation that Ex must satisfy.(b) Show that Ex = E0 cos(?t ? ?z) is a solution of that equation for a
Answer Preview: a First use in which case the curl has dictated the direction that H must lie in Similarly use …

, Chapter: 12 -Problem: 20 >> The 50-MHz plane wave of Problem 12.12 is incident onto the ocean surface at an angle to the normal of 60?. Determine the fractions of the incident power that are reflected and transmitted for(a) s-polarization, and(b) p-polarization.
Answer Preview: a To review Problem 12 we first we find the loss tangent This value is sufficiently grea…

, Chapter: 4 -Problem: 32 >> (a) Using Eq. (35), find the energy stored in the dipole field in the region r > a. (b) Why can we not let a approach zero as a limit? Transcribed Image Text: Qd 4??0r3 + sin ? a ) . (2 cos ? a, E =
Answer Preview: a We start with Then the energy will be b From the above re…

, Chapter: 8 -Problem: 42 >> Find the mutual inductance between two filaments forming circular rings of radii a and a, where a << a. The field should be determined by approximate methods. The rings are coplanar and concentric.
Answer Preview: We use the result of Problem 8 4 which asks for the magne

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