完全版电磁场与电磁波习题解读

2.4 Verify the Pythagorean theorem. In other words, show that 222A B A B +=+ if and only if A is perpendicular toB .2.8 Show that ()()()()()()A B C D A CB D A D BC ⨯⋅⨯=⋅⋅-⋅⋅2.17 Express that position vector ˆˆˆr xxyy zz =++ in the spherical coordinate system. 2.20 ˆˆˆ125S rθπϕ=++ and ˆˆ20.5T r πθ=+ are two vectors at point ()2,,2ππ and ()5,2,2ππ, respectively. Determine (a) S T +,(b) S T ⋅,(c) S T ⨯, (d) the unit vector perpendicular to S T ⨯ and (e) the angle between S andT2.40 Show that (a)sin cos x φφρρφ∂∂∂=-∂∂∂, and (b)cos sin y φφρρφ∂∂∂=+∂∂∂ in the cylindrical coordinate system. 2.41 Show that 22222221x y ρρρρρφ⎡⎤⎛⎫∂∂∂∂∂+=+⎢⎥ ⎪∂∂∂∂∂⎝⎭⎣⎦in the cylindrical coordinate system. 2.42 If the electric field intensity in space is given as 00ˆˆcos sin E E rE θθθ=-, find E ∇⋅ and E ∇⨯. 2.44 Given 322ˆˆˆF x xx yy x zz =++, verify the divergence theorem when the region is bounded by a cylinder 2216x y += and the planes at 0z =and 2z =.2.37 Using the rectangular coordinate system, verify that (a) ()0A ∇⋅∇⨯=and (b) ()0f ∇⨯∇=2.38 In electromagnetic fields, we will define the electric field intensity E as a negative gradient of a scalar function ϕ; i.e.E ϕ=-∇. We will also define the volume charge distribution as 0v E ρε=∇⋅, where 0ε is the permittivity of free space. Determine E and v ρif (a)()0ln V a ϕφρ= in cylindrical coordinates, where 0V and a are constants, (b)0cos V r ϕθ=in spherical coordinates, and 0sin V r ϕθ=in spherical coordinates.2.48 Verify Stokes ’ theorem over a hemispherical surface at 2r =and 02θπ≤≤for a function ˆ100cos F rθ=. 能够给出最后计算结果的，以后都要给出最后计算结果而不是表达式)3.6 An electron is circling around nucleus of a hydrogen atom at a radius of 0.05nm. Determine the angular velocity and the time period of the electron.3.8 The surface charge density on a quarter-disc in the first quadrant is 2cos K C m φ. If the radius of the disc is a, find the E field at a point at ()0,0,P h .3.13 Two straight uniformly charged lines are extending along the z axis from z =-∞to z =+∞. One line carries a charge distribution of 1C m μand is situated at 3y m =-, and the other line is at 3y m =and has a charge distribution of 1C m μ-. Determine the E field at appoint on the x axis at 4x m =.3.14 Three infinite electric sheets carrying uniform charge densities of 2,-3 and 0.52C m μare each separated by a 1-mm air gap, as shown in figure. Find the electric field intensity everywhere in space.3.15 An arc radius 0.2m lies in the 0z =plane and extends from 02φπ≤≤. It has a charge distribution of 600sin 2nC m φ. Determine the E field at (a) a point ()0,0,1P , and (b) the origin.3.16 A finite line extends from 10z m =-to 10z m =and carries a charge distribution of 100znC m . Determine the E field at a point 2 meters away from the line in the 0z =plane.3.25 A spherical volume of radius b supports a charge distribution of 2k r everywhere except at 0r =. Determine the electric flux density in the region (a)r b <, and (b)r b > 5.5 Find the magnetic flux density at point P in Figure 5.5.Figure 5.53.41 A long coaxial cable consists of an inner conductor of radius a and an outer conductor of inner radius b and outer radius c . The inner conductor carries a uniform surface charge distribution ρs . Find the E field at any point in space when (a) the outer conductor is not grounded, and (b) the outer conductor is grounded. (考考大家对静电平衡的感觉)3.42 A uniform charge density ρs exists on the surface on f an infinitely long metal cylinder of radius a . It is surrounded by a concentric dielectric of inner radius a and out radius b . Compute (a) D , E , and P at all points in space, (b) the bound charge densities, and (c) the energy density in the system. （c 部分暂缓）3.43 Find an expression for E ∇⋅in a charge-free but nonhomogeneous medium.3.44 The permittivity of a dielectric medium is given as nz εα=where αand n are constants. If the electric field intensity in the medium has on a z component, show that E nE z ∇⋅=-, where E is the magnitude of the electric field intensity. 3.45 A metal sphere of radius b has uniform surface charge distribution. The permittivity of the surrounding region varies as ()01a r εε=+. Find (a) D, E, and P everywhere in space, (b) the bound charge densities, and (c) the energy density, and (d) show that the potential in the dielectric region is ()0ln 14Q V a r aπε=+, where Q is the total charge on the sphere. (c,d 部分暂缓)5.32 A coaxial cable consists of a long cylindrical conductor of radius a surrounded by a cylindrical shell of inner a radius b and outer radius c . The inner conductor and the outer shell each carry equal and opposite currents I uniformly distribution through the conductors. Obtain expressions for the magnetic field intensity in each of the regions (a)a ρ≤, (b) a b ρ≤≤, (c)b c ρ≤≤, and (d)c ρ≥5.34 A current I flows in a thin wire bent into a circle of radius b. The axis of the circular loop coincides with the z axis. ComputeH dl ⋅⎰along the z axis form z =-∞to z =∞. What do you conclude from your answer?5.35 The magnetic flux density in a region1,0z >, is ˆˆˆ1.50.80.6B x y zmT =++. If 0z =marks the boundary betweenregions 1 and 2, determine the magnetic flux density in region 2. Consider region1 as free space; regions 2 has a relative permeability of 100.5.36 A current sheet of ˆ12ykA m separates two regions at 0z =. Region 2,0z >, has a magnetic field intensity of ˆˆˆ405012xy z kA m ++and a relative permeability of 200. If region 1,0z <, has a permeability of 1000, determine themagnetic field intensity in this region.5.37 Refer to Problem 5.35. Compute the magnetization vector in each region. What the bound volume and surface current densities in each region.由于题目都是类似的，故没有写题目，仅仅画图！ 3.58 Capacitance?Depth=10cm3.59 Capacitance?Area=10cm*10cm0.5mm 0.5mm3.60 Capacitance?(a=1cm, b=1.5cm)1r ε=3.61 Capacitance? (红色为极板)x（xoy 方向俯视图，z 方向高度为h=10cm ）3.62 Capacitance?（球形）半径a=10cm b=20cm c=30cm7.13 Three coils with 100, 150, and 200 turns are tightly and closely wound on a common magnetic circuit of 40cm2 in cross section and 80cm in length. Determine (a) the self-inductance of each coil, (b) the mutual inductance between any twocoils. Assume the relative permeability of the magnetic material as 500.3.63 A large spherical cloud of radius b has a uniform volume charge distribution of ρv . Calculate and sketch the potential distribution and the electric field intensity at any point in space using Poisson’s and Laplace ’s equations.3.66 Determine the potential distribution in a coaxial cable when the inner conductor of radius 10cm is at a potential of 100v and the outer conductor radius 20cm is grounded. The space between the conductors is filled with two concentric dielectrics. The dielectric constant is 3 for the inner dielectric and 9 for the outer dielectric. The dielectric interface is at a radius of 15cm. Determine (a) the potential distribution, (b) the D and E field in each region, (c) the surface charge density on the inner conductor, and (d) the capacitance of the 100-meter-long cable.3.67 Calculate the capacitance of two spherical shells of radii a and b using Laplace ’s equation. The inner shell is at a potential of V0, and the outer shell is grounded. What is the surface charge density on the inner shell? Obtain the expression for the capacitance of the system.3.71 Two infinite grounded metal plates parallel to the z axis are at 00,60φ=. A charge of 500nC is placed at ()05,30,10m. Calculate the potential function and the electric field intensity at ()07,30,10m. Assume that the medium between the plates is free space.7.1 Two conducting bars are sliding over two stationary conductors, as show in figure. What is the induced current in the closed loop thus formed when its resistance is 12Ω?7.29 If the electric field intensity in a source-free, dielectric medium is given as()()0ˆsin sin E E x t x t yVm αωαω=-++⎡⎤⎣⎦, Determine the magnetic field intensity using Maxwell ’s equation from Faraday’s law. What is the displacement currentdensity in the medium? (此题ε和μ应该算已知)7.30 If the magnetic field intensity in a source-free, dielectric medium is given as()()0ˆcos cos H H x t x t zA m αωαω=-++⎡⎤⎣⎦, Determine the electric field intensity using Maxwell ’s equation from Ampere ’s law. What is the displacement currentdensity in the medium?7.31 Determine the condition that must be satisfied for the existence of the electric field intensity as given in Problem 7.29. 7.32 Can the magnetic field intensity as given in Problem 7.30 exist? If yes, what must be the condition? If no, explain why the fields cannot exist.7.33 If ()0ˆcos E E t z xVm ωβ=-in a dielectric medium, show that the electric energy density is equal to the magnetic energy density. Also compute (a) the Poynting vector, (b) the average power density, and (c) the time-average values of theenergy densities. (b 和c 的差别一个是数值上的平均值，一个是对时间的平均值)7.34 Compute the tine-average electric and magnetic energy densities for the fields given in Problem 7.29. 7.35 Compute the tine-average electric and magnetic energy densities for the fields given in Problem 7.307.49 For electromagnetic fields to exist in a linear, homogeneous, isotropic, source-free conductive region, show that the E field must satisfy the following equation:()220E j E ωμεωμσ∇+-=.7.50 For electromagnetic fields to exist in a linear, homogeneous, isotropic, source-free conductive region, show that the H field must satisfy the following equation:()220H j H ωμεωμσ∇+-=7.51 In a dielectric medium (04εε=and 0μμ=) the E and H fields are given as follows:1000cos 3z E t x V m πω⎛⎫=-⎪⎝⎭ and 1000cos 3y H t x A m πωη⎛⎫=--⎪⎝⎭Using phasor analysis, determine (a)ωand η, (b)the direction of power flow, and (c) the average power crossing the surface area bounded by the corners of triangle at (2,0,0)m, (2,4,0)m, and (2,4,2)m. （7.34和7.35算思考题）8.4 Show that the electric field intensity as given by (8ˆ100sin 10E t zVm =+is a valid solution of the wave equation in a dielectric medium. Also determine (a) the electric constant of the medium assuming that 1r μ=, (b) thevelocity of propagation, (c) the magnetic field intensity, (d) the intrinsic impedance, (e) the wavelength, and (f) the average power density.8.6 The electric field intensity of a uniform plane wave in free space is given by ()9ˆ120cos 210E t y zVm πβ=⨯-. Determine (a) the phase constant, (b) the magnetic field intensity, (c) the wavelength, (d) the average power density in the medium, (e) the average energy density in the electric field, and (f) the average energy density in the magnetic field. 8.10 A uniform plane wave at a frequency of 100MHz propagates in the z direction in a conducting medium characterized by 1r μ= 2.25r ε=and 9.375mS m σ=. Determine γ,α,β,δ,p v and tan δ(不会把前面的δ取正切吧，否则晕倒). If the maximum value of the electric field intensity in the y direction is 125V m and occurs at z=0 and t=0, write expressions for the E and H field. Obtain a general expression for the power density in the medium. What is the total current density in the medium?8.11 A glass material is coated with silver (76.110S m σ-=⨯) to make it a good conductor at 2.4GHz. What is the surface resistance of the coating?8.14 An antenna just beneath the surface of a most soil (16r ε=,1r μ=5mS m σ=) is transmitting a signal at 60MHz. Assuming that signal propagates as a uniform plane wave, a receiver buried in the soil receives the signal with a strength of 1/10 of its original value. Compute the distance of the receiver. 8.15Theelectricfield intensityof a wave in a region is givenby ()()0ˆˆ3cos454sin 45E t x yt x zV m ωβωβ=--+-+; determine the polarization of the wave.8.16 Find the polarization of a wave if the electric field intensity is given by ()()0.01120ˆˆ2525j y E j xz e V m +=-+. 8.22 An electromagnetic wave propagates in a dielectric medium with 02.25εε=along the x direction. It strikes another dielectric medium with 09εε=at x=0. If the incoming wave is polarized in the z direction, has a maximum value of250mV/m at the interface, and has an angular frequency of 300Mrad/s, determine (a) the reflection coefficient, (b) the transmission coefficient, and (c) the power densities of the incident, reflected, and transmitted waves.8.21 If we define the standing wave ratio (SWR) as the ratio of the maximum to the minimum values of the total E field in medium 1, show that ())maxmin 11SWR E E ρρ==+- where ρis the magnitude of the reflection coefficient.----------------------------------------------------------------------------------------------------------------------8.17 A uniform plane wave is propagating in a good conductor. If the magnetic field intensity is given by ()158ˆ0.1cos 21015zH et z xA m π-=⨯-, determine the conductivity and the corresponding component of the E field. Calculate the average power loss in a block of unit area and δthickness.8.22 Find the polarization of the following waves: (a) 300300ˆˆ100100j x j x E e y e zV m --=+ (b) 41004100ˆˆ169j j zj j z E ee xe e yV m ππ----=- (c) ()()ˆˆ3cos 0.54sin 0.5E t y xt y zV m =--- 8.31 A uniform plane wave is traveling from one dielectric medium into another dielectric medium. The dielectric constant of one medium is 1.25. If the permeability of each medium is the same as that of free space, and the reflection coefficient is 0.25, determine the dielectric constant of the other medium.8.33 A uniform plane wave propagating in free space strikes the plane surface of a dielectric ( 2.25r ε=,1r μ=) at an angle θwith respect to the normal. If the interface is at z=0, and the electric field intensity of the incident waveis ()6ˆ50cos 3100.7660.643t z y xVm ⨯-+, determine θ, the phase velocity, and the group velocity. Write expressions for the incident, reflected, and transmitted waves. Compute the power density in each region.今天才注意到，这本书中作业题分为两个部分，一个是exercises ，一个是problems ，这次两个部分都出了一点，感觉exercises 比较简单，而problems 比较难（以往出的题目都是这一部分的），快考试了，你们愿意出那一部分的题目。