Engineering Electromagnetics 8th edition William H. Hayt, John A.Buck - Solutions

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 definition in Problem 14.14;(c) Determine the maximum directivity.
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 intensity. Express your result as the sidelobe level in decibels, given by Ss[dB] = 10
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 = 30◦).(b) Find Eθs at P(100, 90◦, 30◦) if the vertical element is located at A(0.1, 90◦,
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 radiation-field amplitude of 100 mV/m at a distance of 1 mi, at θ = 90◦;(b) Radiate a total power
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) Provide a radiation-field amplitude of 100 mV/m at a distance of 1 mi, at θ = 90◦;(b) Radiate 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 − |z|)/0.5d;(c) Step, I0 for 0 < |z| < 0.25d and 0.5I0 for 0.25d < |z| < 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 magnetic field is the same as that found by applying the Biot€“Savart law (Eq. (2), Chapter 7)
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 equations survives, and the phases, Î´rand Î´Î¸, simplify to a single
Two short antennas at the origin in free space carry identical currents of 5 cos ωt A, one in the az direction, and one in the ay direction. Let λ = 2π m and d = 0.1 m. Find Es at the distant point where(a) (x = 0, y = 1000, z = 0);(b) (0, 0, 1000);(c) (1000, 0, 0).(d) Find E at (1000, 0, 0) at
Prepare a curve, r vs. θ in polar coordinates, showing the locus in the ϕ = 0 plane where(a) The radiation field |Eθs| is one-half of its value at r = 104 m, θ = π/2;(b) Average radiated power density r> is one-half its value at r = 104 m, θ = π/2.
A short dipole-carrying current I0 cos ωt in the az direction is located at the origin in free space.(a) If k = 1 rad/m, r = 2 m, θ = 45◦, ϕ = 0, and t = 0, give a unit vector in rectangular components that shows the instantaneous direction of E.(b) What fraction of the total average power is
The mode field radius of a step index fiber is measured as 4.5 μm at free-space wavelength λ = 1.30 μm. If the cutoff wavelength is specified as λc = 1.20 μm, find the expected mode field radius at λ = 1.55 μm.
Is the mode field radius greater than or less than the fiber core radius in single-mode step index fiber?
A step index optical fiber is known to be single mode at wavelengths λ > 1.2μm. Another fiber is to be fabricated from the same materials, but it is to be single mode at wavelengths λ > 0.63μm. By what percentage must the core radius of the new fiber differ from the old one, and should it
An asymmetric slab waveguide is shown in Figure 13.26. In this case, the regions above and below the slab have unequal refractive indices, where n1> n3> n2.(a) Write, in terms of the appropriate indices, an expression for the minimum possible wave angle, Î¸1, that a guided mode
In a symmetric slab waveguide, n1 = 1.50, n2 = 1.45, and d = 10μm.(a) What is the phase velocity of the m = 1 TE or TM mode at cutoff?(b) How will your part (a) result change for higher-order modes (if at all)?
A symmetric slab waveguide is known to support only a single pair of TE and TM modes at wavelength λ = 1.55μm. If the slab thickness is 5 μm, what is the maximum value of n1 if n2 = 3.30?
Consider a transform-limited pulse of center frequency f = 10 GHz, and of full-width 2T = 1.0 ns. The pulse propagates in a lossless single-mode rectangular guide which is air-filled and in which the 10 GHz operating frequency is 1.1 times the cutoff frequency of the TE10mode. Using the result of $d²B We\27-3/2 do?$
Show that the group dispersion parameter, d2Î²/dÏ‰2, for a given mode in a parallel-plate or rectangular waveguide is given bywhere Ï‰c is the radian cutoff frequency for the mode in question [note that the first derivative form was already found, resulting in Eq. $d²B We\27-3/2 do?$
Integrate the result of Problem 13.22 over the guide cross section, 0 < x < a, 0 < y < b, to show that the average power in watts transmitted down the guide is given aswhere Î· = ˆšÎ¼/ˆˆ and Î¸10 is the wave angle associated with
Using the relation (S) = 1/2 Re{EsÃ— Hˆ—s} and Eqs. (106) through (108), show that the average power density in the TE10 mode in a rectangular waveguide is given by
An air-filled rectangular waveguide is to be constructed for single-mode operation at 15 GHz. Specify the guide dimensions, a and b, such that the design frequency is 10 percent higher than the cutoff frequency for the TE10 mode, while being 10 percent lower than the cutoff frequency for the
Two rectangular waveguides are joined end-to-end. The guides have identical dimensions, where a = 2b. One guide is air-filled; the other is filled with a lossless dielectric characterized by ∈'r.(a) Determine the maximum allowable value of r such that single-mode operation can be simultaneously
A rectangular waveguide has dimensions a = 6 cm and b = 4 cm.(a) Over what range of frequencies will the guide operate single mode?(b) Over what frequency range will the guide support both TE10 and TE01 modes and no others?
In the guide of Figure 13.25, it is found that m = 1 modes propagating from left to right totally reflect at the interface, so that no power is transmitted into the region of dielectric constant ˆˆ'r2.(a) Determine the range of frequencies over which this will occur.(b) Does your part (a)
A parallel-plate guide is partially filled with two lossless dielectrics (Figure 13.25) where ˆˆ'r1= 4.0, ˆˆ'r2= 2.1, and d = 1 cm. At a certain frequency, it is found that the TM1mode propagates through the guide without suffering any reflective loss at the dielectric
The cutoff frequency of the m = 1 TE and TM modes in an air-filled parallel-plate guide is known to be fc1 = 7.5 GHz. The guide is used at wavelength λ = 1.5 cm. Find the group velocity of the m = 2 TE and TM modes.
For the guide of Problem 13.14, and at the 32 GHz frequency, determine the difference between the group delays of the highest-order mode (TE or TM) and the TEM mode. Assume a propagation distance of 10 cm.In ProblemA d = 1 cmparallel-plate guide is made with glass (n = 1.45) between plates. If the
A d = 1 cmparallel-plate guide is made with glass (n = 1.45) between plates. If the operating frequency is 32 GHz, which modes will propagate?
A lossless parallel-plate waveguide is known to propagate the m = 2 TE and TM modes at frequencies as low as 10 GHz. If the plate separation is 1 cm, determine the dielectric constant of the medium between plates.
A parallel-plate guide is to be constructed for operation in the TEM mode only over the frequency range 0 < f < 3 GHz. The dielectric between plates is to be teflon (∈'r = 2.1). Determine the maximum allowable plate separation, d.
A parallel-plate waveguide is known to have a cutoff wavelength for the m = 1 TE and TM modes of λc1 = 4.1 mm. The guide is operated at wavelength λ = 1.0 mm. How many modes propagate?
Two micro-strip lines are fabricated end-to-end on a 2-mm-thick wafer of lithium niobate (∈'r = 4.8). Line 1 is of 4 mm width; line 2 (unfortunately) has been fabricated with a 5 mm width. Determine the power loss in dB for waves transmitted through the junction.
A microstrip line is to be constructed using a lossless dielectric for which ∈'r = 7.0. If the line is to have a 50 Ω characteristic impedance, determine(a) ∈r,eff;(b) w/d.
A transmission line constructed from perfect conductors and an air dielectric is to have a maximum dimension of 8 mm for its cross section. The line is to be used at high frequencies. Specify the dimensions if it is(a) A two-wire line with Z0 = 300 Ω;(b) A planar line with Z0 = 15 Ω;(c) A 72 Ω
Pertinent dimensions for the transmission line shown in Figure 13.2 are b = 3 mm and d = 0.2 mm. The conductors and the dielectric are nonmagnetic.(a) If the characteristic impedance of the line is 15 Î©, find ˆˆ'r. Assume a low-loss dielectric.(b) Assume copper conductors and
The transmission line in Fig. 6.8 is filled with polyethylene. If it were filled with air, the capacitance would be 57.6 pF/m. Assuming that the line is lossless, find C, L, and Z0.
Each conductor of a two-wire transmission line has a radius of 0.5 mm; their center-to-center separation is 0.8 cm. Let f = 150 MHz, and assume σ and σc are zero. Find the dielectric constant of the insulating medium if(a) Z0 = 300 Ω;(b) C = 20 pF/m;(c) νp = 2.6 × 108 m/s.
Find R, L, C, and G for a two-wire transmission line in polyethylene at f = 800 MHz. Assume copper conductors of radius 0.50 mm and separation 0.80 cm. Use ∈'r = 2.26 and σ/(ω∈') = 4.0 × 10−4.
Two aluminum-clad steel conductors are used to construct a two-wire transmission line. Let σAl = 3.8 × 107 S/m, σSt = 5 × 106 S/m, and μSt = 100μH/m. The radius of the steel wire is 0.5 in., and the aluminum coating is 0.05 in. thick. The dielectric is air, and the center-to-center wire
Find R, L, C, and G for a coaxial cable with a = 0.25 mm, b = 2.50 mm, c = 3.30 mm, ∈'r = 2.0, μr = 1, σc = 1.0 × 107 S/m, σ = 1.0 × 10−5 S/m, and f = 300 MHz.
The conductors of a coaxial transmission line are copper (σc = 5.8 × 107 S/m), and the dielectric is polyethylene (∈'r = 2.26, σ/ω∈' = 0.0002). If the inner radius of the outer conductor is 4 mm, find the radius of the inner conductor so that(a) Z0 = 50 Ω;(b) C = 100 pF/m;(c) L = 0.2μH/m.
A T = 20 ps transform-limited pulse propagates through 10 km of a dispersive medium for which β2 = 12 ps2/km. The pulse then propagates through a second 10 km medium for which β2 = −12 ps2/km. Describe the pulse at the output of the second medium and give a physical explanation for what
A T = 5 ps transform-limited pulse propagates in a dispersive medium for which β2 = 10 ps2/km. Over what distance will the pulse spread to twice its initial width?
Over a small wavelength range, the refractive index of a certain material varies approximately linearly with wavelength as n(λ) = na + nb(λ − λa), where na, nb and λa are constants, and where λ is the free-space wavelength.(a) Show that d/dω = −(2πc/ω2)d/dλ.(b) Using β(λ) = 2πn/λ,
Using Eq. (79) in Chapter 11 as a starting point, determine the ratio of the group and phase velocities of an electromagnetic wave in a good conductor. Assume conductivity does not vary with frequency.Eq. (79)α = β = √πfμσ
Show how a single block of glass can be used to turn a p-polarized beam of light through 180◦, with the light suffering (in principle) zero reflective loss. The light is incident from air, and the returning beam (also in air) may be displaced sideways from the incident beam. Specify all pertinent
In the Brewster prism of Figure 12.18, determine for s-polarized light the fraction of the incident power that is transmitted through the prism, and from this specify the dB insertion loss, defined as 10log10of that number. п
A Brewster prism is designed to pass p-polarized light without any reflective loss. The prism of Figure 12.18 is made of glass (n = 1.45) and is in air. Considering the light path shown, determine the vertex angle Î±. п
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
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.
You are given four slabs of lossless dielectric, all with the same intrinsic impedance, η, known to be different from that of free space. The thickness of each slab is λ/4, where λ is the wavelength as measured in the slab material. The slabs are to be positioned parallel to one another, and the
A uniform plane wave in free space is normally incident onto a dense dielectric plate of thickness λ/4, having refractive index n. Find the required value of n such that exactly half the incident power is reflected (and half transmitted). Remember that n > 1.
A left-circularly polarized plane wave is normally incident onto the surface of a perfect conductor.(a) Construct the superposition of the incident and reflected waves in phasor form.(b) Determine the real instantaneous form of the result of part (a).(c) Describe the wave that is formed.
A right-circularly polarized plane wave is normally incident from air onto a semi-infinite slab of plexiglas (∈'r = 3.45, ∈"r = 0). Calculate the fractions of the incident power that are reflected and transmitted. Also, describe the polarizations of the reflected and transmitted waves.
A 50-MHz uniform plane wave is normally incident from air onto the surface of a calm ocean. For seawater, σ = 4 S/m, and ∈'r = 78.(a) Determine the fractions of the incident power that are reflected and transmitted.(b) Qualitatively, how (if at all) will these answers change as the frequency is
A 150-MHz uniform plane wave is normally incident from air onto a material whose intrinsic impedance is unknown. Measurements yield a standing wave ratio of 3 and the appearance of an electric field minimum at 0.3 wavelengths in front of the interface. Determine the impedance of the unknown
In Figure 12.1, let region 2 be free space, while Î¼r1= 1, ˆˆ"r1= 0, and ˆˆ'r1is unknown. Find ˆˆ'r1if(a) The amplitude of Eˆ’1 is one-half that of E+1;(b) (Sˆ’1) is one-half of (S+1)(c) |E1|min is one-half of |E1|max. Region 2 Иу
Region 1, z < 0, and region 2, z > 0, are both perfect dielectrics (μ = μ0, = 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 λ1 = 5 cm and λ2 = 3 cm. What percentage of the energy incident on the
A wave starts at point a, propagates 1 m through a lossy dielectric rated at 0.1 dB/cm, reflects at normal incidence at a boundary at which Γ = 0.3 + j0.4, and then returns to point a. Calculate the ratio of the final power to the incident power after this round trip, and specify the overall loss
The semi-infinite regions z < 0 and z > 1 m are free space. For 0 < z < 1 m, ∈'r = 4, μr = 1, and r = 0. A uniform plane wave with ω = 4 × 108 rad/s is traveling in the az direction toward the interface at z = 0.(a) Find the standing wave ratio in each of the three regions.(b) Find
In the beam-steering prism of Example 12.8, suppose the antireflective coatings are removed, leaving bare glass-to-air interfaces. Calcluate the ratio of the prism output power to the input power, assuming a single transit.
The region z < 0 is characterized by ∈'r = μr = 1 and r = 0. The total E field here is given as the sum of two uniform plane waves, Es = 150 e−j10zax + (50 ∠ 20◦) ej10zax V/m.(a) What is the operating frequency?(b) Specify the intrinsic impedance of the region z > 0 that would
A 10 MHz uniform plane wave having an initial average power density of 5 W/m2 is normally incident from free space onto the surface of a lossy material in which ∈"2/∈'2 = 0.05, ∈'r2 = 5, and μ2 = μ0. Calculate the distance into the lossy medium at which the transmitted wave power density is
A uniform plane wave in region 1 is normally incident on the planar boundary separating regions 1 and 2. If ∈"1 = ∈"2 = 0, while ∈'r1 = μ3r1 and ∈'r2 = μ3r2, find the ratio ∈'r2/∈'r1 if 20% of the energy in the incident wave is reflected at the boundary. There are two possible
The plane z = 0 defines the boundary between two dielectrics. For z < 0, ∈r1 = 9, ∈"r1 = 0, and μ1 = μ0. For z > 0, ∈'r2 = 3, ∈"r2 = 0, and μ2 = μ0. Let E+x1 = 10 cos(ωt − 15z) V/m and find(a) ω;(b) (S+1);(c) (S−1);(d) (S+2).
Given a general elliptically polarized wave as per Eq. (93):Es = [Ex0ax + Ey0ejϕay ]e−jβz(a) Show, using methods similar to those of Example 11.7, that a linearly polarized wave results when superimposing the given field and a phase-shifted field of the form:Es = [Ex0ax + Ey0e−jϕay
Given a wave for which Es = 15e−jβzax + 18e−jβzejϕay V/m in a medium characterized by complex intrinsic impedance, η(a) Find Hs;(b) Determine the average power density in W/m2.
A linearly polarized uniform plane wave, propagating in the forward z direction, is input to a lossless anisotropic material, in which the dielectric constant encountered by waves polarized along y(∈ry) differs from that seen by waves polarized along x(∈rx). Suppose ∈rx = 2.15, ∈ry = 2.10,
In an anisotropic medium, permittivity varies with electric field direction, and is a property seen in most crystals. Consider a uniform plane wave propagating in the z direction in such a medium, and which enters the material with equal field components along the x and y axes. The field phasor
Consider a left circularly polarized wave in free space that propagates in the forward z direction. The electric field is given by the appropriate form of Eq. (100). Determine(a) The magnetic field phasor, Hs;(b) An expression for the average power density in the wave in W/m2 by direct application
A uniform plane wave in free space has electric field vector given by Es = 10e−jβxaz + 15e−jβxay V/m.(a) Describe the wave polarization.(b) Find Hs.(c) Determine the average power density in the wave in W/m2.
The planar surface z = 0 is a brass-Teflon interface. Use data available in Appendix C to evaluate the following ratios for a uniform plane wave having ω = 4 × 1010 rad/s:(a) αTef/αbrass;(b) λTef/λbrass;(c) vTef/νbrass.
The dimensions of a certain coaxial transmission line are a = 0.8 mm and b = 4 mm. The outer conductor thickness is 0.6 mm, and all conductors have σ = 1.6 × 107 S/m.(a) Find R, the resistance per unit length at an operating frequency of 2.4 GHz.(b) Use information from Sections 6.3 and 8.10 to
A good conductor is planar in form, and it carries a uniform plane wave that has a wavelength of 0.3 mm and a velocity of 3 × 105 m/s. Assuming the conductor is nonmagnetic, determine the frequency and the conductivity.
A hollow tubular conductor is constructed from a type of brass having a conductivity of 1.2 × 107 S/m. The inner and outer radii are 9 and 10 mm, respectively. Calculate the resistance per meter length at a frequency of(a) dc;(b) 20 MHz;(c) 2 GHz.
The inner and outer dimensions of a coaxial copper transmission line are 2 and 7 mm, respectively. Both conductors have thicknesses much greater than δ. The dielectric is lossless and the operating frequency is 400 MHz. Calculate the resistance per meter length of the(a) Inner conductor;(b) Outer
Consider the power dissipation term, ʃ E · Jdv, in Poynting’s theorem (Eq. (70)). This gives the power lost to heat within a volume into which electromagnetic waves enter. The term pd = E · J is thus the power dissipation per unit volume in W/m3. Following the same reasoning that resulted in
Let η = 250 + j30Ω and jk = 0.2 + j2m−1 for a uniform plane wave propagating in the az direction in a dielectric having some finite conductivity. If |Es| = 400 V/m at z = 0, find(a) (S) at z = 0 and z = 60 cm;(b) The average ohmic power dissipation in watts per cubic meter at z = 60 cm.
Given a 100-MHz uniform plane wave in a medium known to be a good dielectric, the phasor electric field is Es = 4e−0.5ze−j20zax V/m. Determine(a) ∈';(b) ∈";(c) η;(d) Hs;(e) S;(f) The power in watts that is incident on a rectangular surface measuring 20 m × 30 m at z = 10 m.
Voltage breakdown in air at standard temperature and pressure occurs at an electric field strength of approximately 3 × 106 V/m. This becomes an issue in some high-power optical experiments, in which tight focusing of light may be necessary. Estimate the light-wave power in watts that can be
The cylindrical shell, 1 cm < ρ < 1.2 cm, is composed of a conducting material for which σ = 106 S/m. The external and internal regions are nonconducting. Let Hϕ = 2000 A/m at ρ = 1.2 cm. Find(a) H everywhere;(b) E everywhere;(c) S everywhere.
In a medium characterized by intrinsic impedance η = |η|ejϕ, a linearly polarized plane wave propagates, with magnetic field given as Hs = (H0yay + H0zaz)e−αx e−jβx. Find(a) Es;(b) E(x, t);(c) H(x, t);(d) S.
Let jk = 0.2 + j1.5 m−1 and η = 450 + j60Ω for a uniform plane propagating in the az direction. If ω = 300 Mrad/s, find μ, ∈', and ∈" for the medium.
A certain nonmagnetic material has the material constants ∈'r = 2 and ∈"/∈' = 4 × 10−4 at ω = 1.5 Grad/s. Find the distance a uniform plane wave can propagate through the material before(a) It is attenuated by 1 Np;(b) The power level is reduced by one-half;(c) The phase shifts 360◦.
A 10 GHz radar signal may be represented as a uniform plane wave in a sufficiently small region. Calculate the wavelength in centimeters and the attenuation in nepers per meter if the wave is propagating in a nonmagnetic material for which(a) ∈'r = 1 and ∈"r = 0;(b) ∈'r = 1.04 and ∈"r =
Uniform current sheets are located in free space as follows: 8az A/m at y = 0, −4az A/m at y = 1, and −4az A/m at y = −1. Find the vector force per meter length exerted on a current filament carrying 7 mA in the aL direction if the filament is located at(a) x = 0, y = 0.5, and aL = az;(b) y =
Two conducting strips, having infinite length in the z direction, lie in the xz plane. One occupies the region d/2 < x < b + d/2 and carries surface current density K = K0az; the other is situated at −(b + d/2) < x < −d/2 and carries surface current density −K0az.(a) Find the
A current of −100az A/m flows on the conducting cylinder ρ = 5 mm, and +500az A/m is present on the conducting cylinder ρ = 1 mm. Find the magnitude of the total force per meter length that is acting to split the outer cylinder apart along its length.
(a) Use Eq. (14), Section 8.3, to show that the force of attraction per unit length between two filamentary conductors in free space with currents I1az at x = 0, y = d/2, and I2az at x = 0, y = −d/2, is μ0 I1 I2/(2πd).(b) Show how a simpler method can be used to check your result.
Two circular wire rings are parallel to each other, share the same axis, are of radius a, and are separated by distance d, where d << a. Each ring carries current I. Find the approximate force of attraction and indicate the relative orientations of the currents.
A current of 6 A flows from M(2, 0, 5) to N(5, 0, 5) in a straight, solid conductor in free space. An infinite current filament lies along the z axis and carries 50 A in the az direction. Compute the vector torque on the wire segment using an origin at:(a) (0, 0, 5);(b) (0, 0, 0);(c) (3, 0, 0).
A solenoid is 25 cm long, 3 cm in diameter, and carries 4 A dc in its 400 turns. Its axis is perpendicular to a uniform magnetic field of 0.8 Wb/m2 in air. Using an origin at the center of the solenoid, calculate the torque acting on it.
A solid conducting filament extends from x = −b to x = b along the line y = 2, z = 0. This filament carries a current of 3 A in the ax direction. An infinite filament on the z axis carries 5 A in the az direction. Obtain an expression for the torque exerted on the finite conductor about an origin
Assume that an electron is describing a circular orbit of radius a about a positively charged nucleus.(a) By selecting an appropriate current and area, show that the equivalent orbital dipole moment is ea2ω/2, where ω is the electron’s angular velocity.(b) Show that the torque produced by a
The hydrogen atom described in Problem 8.16 is now subjected to a magnetic field having the same direction as that of the atom. Show that the forces caused by B result in a decrease of the angular velocity by eB/(2me) and a decrease in the orbital moment by e2a2B/(4me). What are these decreases for
Calculate the vector torque on the square loop shown in Figure 8.15 about an origin at A in the field B, given(a) A(0, 0, 0) and B = 100ay mT;(b) A(0, 0, 0) and B = 200ax + 100ay mT;(c) A(1, 2, 3) and B = 200ax + 100ay ˆ’ 300az mT;(d) A(1, 2, 3) and B = 200ax + 100ay ˆ’ 300az mT

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Engineering Electromagnetics 8th edition William H. Hayt, John A.Buck - Solutions

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