1. A line charge of uniform density ρl in free space forms a semicircle of radius b. Determine the magnitude and direction of the electric field intensity at the center of the semicircle.[高考]
  2. Determine the electric field caused by spherical cloud of electrons with a volume charge densityρ=-ρ0 for0Rb(bothρ0 and b are positive) andρ=0 for Rb.[交大電子物理所]
  3. At what value of θ does the electric field intensity of a z-directed dipole have no z-component.
  4. Obtain a formula for electrical field intensity along the axis of a uniform line charge of length L. The uniform line-charge density isρl. [高考]
  5. A finite line charge of length L=3m carrying uniform line charge density is coincident with the x-axis. Determine V and in the plane bisecting the line charge.
  6. A charge distributed uniform over an L×L square plate. Determine V and at a point on the axis perpendicular to the plate and through its center.
  7. A current I flows in the inner conductor of an infinitely long coaxial line and returns via the outer conductor. The radius of the inner conductor is a, and the inner and outer radii of the outer conductor are b and c, respectively. Find the magnetic flux densityfor all regions and plotversus r.[高考]
  8. A direct current I=1A flows in a straight wire of length 2L=8m. Find the magnetic flux density at a point located at a distance r=5m from the wire in the bisecting plane.
  9. Find the magnetic flux density at the center of a square loop, with side w=1m carrying a direct current I=3A.[台大物理研究所]
  10. Find the magnetic flux density at a point on the axis of a circular loop of radius b that a direct current I.
  11. Determine the magnetic flux density at a point on the axis of a uniformly magnetized circular cylinder of magnetic material. The cylinder has a radius b=1m, length L=3m, and axial magnetization (A/m).[交大電子研究所]
  12. Determine the magnetic flux density at a point on the axis of a solenoid of radius b=1m and length L=3m, and with a current I=1A in its 1000 turns of closely wound coil.
  13. The figure shows an infinitely long solenoid with air core having n closely wound turns per unit length. The windings are slanted at an angle α and carry a current I. Determine the magnetic flux density both inside and outside the solenoid.
  14. In certain experiments it is desirable to have a region of constant magnetic flux density. This can be created in an off-center cylindrical cavity. The uniform axial current density is . Find the magnetic and direction of in the cylindrical cavity whose axis is displaced from that of the conducting part by a distance d.[台大電研、清大電研、中原電機]
  15. A conducting sliding bar oscillates over two parallel conducting rails of length 0.7m in a sinusoidally varying magnetic field (mT). The distance of the two rails and the position of the sliding bar are respectively given by d=0.2m and (m), and the rails are terminated in a resistance R=0.2Ω. Find the current ion the rails.
  16. What is the Hall effect?
  17. An electron is injected with an initial velocity into a region where both an electric field and a magnetic field exist. Describe the motion of the electron if and .
  18. Determine the force per unit length between two infinitely long parallel conducting wires carrying currents I1=1A and I2=3A in the opposite direction. The wires are separated by a distance d=2m. [高考]
  19. Two dielectric media are separated by a charge free boundary as show in figure. The electric field intensity in media 1 at the point P1 has a magnitude E1 and makes an angle with the normal. Determine the magnitude E2 and direction of the electric field intensity at point P2 in medium 2.[交大電子所]
  20. A lucite sheet (εr=3.2) is introduced perpendicularly in a uniform electric field in free space. Determine ,, and inside the lucite. [中央地球物理所]
  21. Assume that the z=0 plane separates two lossless dielectric regions withεr1=2 andεr2=3. If in region 1, what about is in region 2 on the xy-plane.
  22. A very large slab of material of thickness d lies perpendicularly to uniform magnetic field intensity . Ignoring edge effect, determine the magnetic field intensity in the slab: (a) if the slab material has a permeability μ, (b) if the slab permanent magnet having a magnetization vector.[台大物理所]
  23. Consider the xy-plane boundary (z=0) between region 1 (μr1=1) and region 2 (μr2=5000). Assuming , find on the boundary.
  24. A point charge Q is at the center of a spherical dielectric shell with inner and outer radii ri and ro. Determine , V, , and in terms of the radial distancer.
  25. According to Gauss‘s law, show that the induced surface charge densityρs=εEn (in which, Enis the normal component of on the surface).
  26. A long carrying a current I folds back with semicircular bend of radius b. Determine magnetic flux density at the center points P of the bend.
  27. A circular rod of magnetic material with permeabilityμis inserted coaxially in the long solenoid. The radius of the rod a is less than the inner radius b of the solenoid. The solenoid’s winding has n turns per unit length and carries a current I. (a) Find the values of,, and inside the solenoid for ra and for arb. (b) What are the equivalent magnetization current densitiesJmand Jms for the magnetized rod?[清大電研]
  28. A ferromagnetic sphere of radius b is magnetized uniformly with a magnetization Determine the equivalent magnetization current densities and . And determine the magnetic flux density at the center of the sphere. [台大電研]
  29. Derive the continuity equation .
  30. An infinite time-harmonic surface current sheet with surface current density Js=located on the z=0 plane. Find the EM wave it produces everywhere in space. (交大電信所)
  31. What is the Lorentz condition?
  32. Given that in air, find and β.
  33. Given that in air, find andβ.
  34. Write down the electromagnetic boundary conditions.
  35. The electric field of an electromagnetic wave is the sum of and . Find E0 and θ.
  36. An elliptically polarized wave travels in z-direction in air with electric field components Ex=2cos(ωt-βz)V/m, Ex=4√2cos(ωt-βz-45°) V/m. Find the average power per square meter area conveyed by the wave. [台大電研]
  37. For each of the following sinusoidal traveling waves, determine if the polarization isright- or left-hand circular or elliptical.

(a)

(b)

(c)[台大電研]

  1. Write down the expression for the electric- and magnetic- field intensities, and , of sinusoidal time-varying uniform plane wave propagating in free space and having the following characteristics: (1) f=10GHz; (2) direction of propagation is the +z direction; (3) left-hand circular polarization; (4) the initial condition is the electric field in the z=0 plane and t=0 having an x-component equal to E0anda y-component equal to√3E0.[台大電研]
  2. A plane 300 MHz traveling wave in air with E(peak)=1V/m is incident normally on a large conductor sheet with σ=6×103Ω-1m-1. Find the average power absorbed by the sheet per square meter area.[台大電研]
  3. Show that the attenuation constant is equal to the propagation constant for a good conductor. That is, . (Hint: ).
  4. Show that the reflection coefficientsΓare equal in both the perpendicular and parallel polarizations for normal incident EM waves, so as the transmission coefficients τ.
  5. A plane wave is incident from air onto a dielectric plate (n=1.5) at an angle of incidence 45°. Determine the intensity reflectances of TE and TM waves. What is the average reflectance for unpolarized incident wave (that carrying TE and TM waves of equal intensities)?[台大電研]
  6. Show that Γ=.
  7. Show that at the Brewster angle of incidence, the directions of the refracted and refracted waves are orthogonal. The electric field of the refracted TM wave is then parallel to the direction of the reflected wave.[台大電研]
  8. Consider the following uniform plane wave in air with the following phasor expression for electric intensity (mV/m) (a) What is the frequency? (b) What is the direction of propagation and the state of polarization? (c) Find the associated magnetic field.[成大電研]
  9. A uniform plane wave in air with the following phasor expression for electric intensity (V/m) is incident on a conducting plane at z=0. (a) Find the frequency and wavelength of the wave. (b) Find and of the reflected wave.
  10. When a plane wave in medium 1 (ε1,μ0) is normally incident on a plane of medium 2 (ε2,μ0), the reflection coefficient isΓ=-1/3. (a) Find the relation betweenε1 andε2. (b) Calculate the Brewster angle for the parallel polarization. (c) Calculate the reflection coefficient for the perpendicular polarization in case of the incident angleθi=30°. [台大電研]
  11. Show that the numerical aperture (N.A.) of a fiber with the refractive index of the core n1 and that of the cladding n2 (n1n2) is .[文化電機期末考]
  12. A coaxial line has inner and outer radii ri=5mm and ro=10mm. The potential difference between them is maintained at V0=2volts. Determine the potential distribution between the inner and outer conductors.
  13. Determine the capacitance per unit length of two conducting wires of radius a=5mm, axis-separation D=15mm, and surrounded by a dielectric with permittivityε=2ε0.
  14. A constant voltage V0=100volts is applied to a parallel-plate capacitor of which separation is d=15mm. The permittivity and thickness of the dielectricA areεA=12ε0 and d1=10mm, respectively. The permittivity and thickness of the dielectricB areεB=6ε0 and d2=5mm, respectively. And the area is S=1m2. Find the, , and capacitance of the capacitor.[台大電研]
  15. Find the energy required to assemble a uniform charge of radius b and volume charge densityρ. [清大電研]
  16. A parallel-plate capacitor of width w, length L, and separation d is partially filled with a dielectric medium of dielectric constants εr. A battery of Vo volts is connected between the plates. Find, , ρs in each region, and find distance x such that the electrostatic energy stored in each region is the same.[台大電研]
  17. A parallel-plate capacitor of length L=2m in the x-direction, width w=1m in the y-direction, and separation d=0.5m in the z-direction, has a dielectric medium ofεr=4 in the space between the plates. The capacitor is charged to a voltage V0=5volts. Assuming the dielectric slab is withdrawn from the capacitor along the x-direction, determine the force acting on the slab under (a) with the switch is closed, (b) after the switch is first opened.[台大、清大電研]
  18. The two plates of a parallel-plate capacitor are separated by a distance d=1m and maintained at potentials 0 and Vo=3volts. Assume negligible fringing effect at the edges, determine (a) the potential at any point between the plates, (b) the surface charge densities on the plates. [清大電研]
  19. The conductors of an isolated two-wire transmission line, each of radius b=1m, are spaced at a distance D=100m apart. Assuming a voltage V0=5volts between the lines, find the force per unit length on the lines.
  20. One end of a coaxial line with inner and outer radii ri=5mm and ro=10mm is short-circuited by a thin, tight-fitting conducting washer. Find the magnitude of the magnetic force on the washer when a current I=2A flows in the line.
  21. There are two electric dipoles above the grounded xy-plane. One is +z-directed and another is +x-directed. Which directions are their respective images pointed at? Give your reasons.[台大電研]
  22. A constant voltage V0=5volts is applied to a partially filled parallel-plate capacitor of which separation is d=1m. The permittivity and thickness of the dielectric areε=2ε0 and d1=0.5m, respectively. And the area is S=1m2. Find the force on the upper plate.
  23. Consider the region enclosed on three sides by grounded conducting planes. The end plate on the left is insulated from the grounded sides and has a constant potential V0. All planes are assumed to be infinite in extent in the z-direction. Determine the potential distribution within this region.[交大電信所]
  24. To insert a z-directional circular conducting cylinder in an original uniform electric field , draw the electric lines before and after placing the cylinder.
  25. Referring to the previous problem, describe what isSt. Elmo’s fire? If you want to take a photo of St. Elmo’s fire, what equipments are needed?
  26. Solve the electric potential in the cylindical coordinate system, the general solution is V(r,φ)=, where Vn(r,φ)=rn(Asinnφ+Bcosnφ)+r-n(A’sinnφ+B’cosnφ), n≠0. But we solve the potential in the left figure, it is often assumed that as rband as rb. Explain the reason.
  27. The potential is the solution of the boundary problems of static fields in the spherical coordinates. However, an uncharged conducting sphere of radiusb is placed in an initially uniform electric field . The potential has only one term in the positive powers ofR. Why?
  28. The space between a parallel-plate capacitor of area S=1m2 is filled with a dielectric whose permittivity varies linearly fromε=ε0 at one plate (y=0) toε=2ε0 at the other plate (y=d=1m). Neglecting fringing effect, find the capacitance.[台科大電子研究所]
  29. The length of a coaxial line with inner and outer radii ri=5mm and ro=10mm is1m. Two dielectric media ofεr1=2 andεr2=3 fill equally the space between the inner and outer conductors. Determine its capacitance.[台大物理所、高考電機技師]
  30. Show that C/G=ε/σ.[台大電研]
  31. An infinite conducting cone of half angle α=30°is maintained at potential V0=5volts and insulated from a grounded conducting plane. Determine (a) the potential distribution V(θ) in the regionαθ<π/2, (b) the electric field intensity in the regionαθ<π/2, (c) the charge densities on the cone surface and on the grounded plane.
  32. Two infinite insulated conducting planes maintained at potentials 0 and V0=5volts form a wedge-shaped configuration with an angleα=π/6. Determine the potential distributions for the regions: (a) 0<φα, and (b) αφ2π. [中央光電所、成大電研]
  33. Assume a rectangular conducting sheet of conductivityσ, width a, and height b. A potential difference is applied to the side edges. Find the potential distribution, and the current density everywhere within the sheet.[台科大電子所]
  34. The space between two parallel conducting plates each having an area S is filled with an inhomogeneous ohmic medium whose conductivity varies linearly fromσ1 at one plate (y=0) toσ2 at the other plate (y=d). And d-c voltage V0 is applied across the plates. Determinethe total resistance between the plates, and the surface charge densities on the plates, and the volume charge density between the plates.
  35. Determine the resistance between two concentric spherical surfaces of radii R1=1m and R2=2m, which is filled with a material of conductivityσ=1×107(S/m). (高考)
  36. Determine the resistance between two concentric spherical surfaces of radii R1=1m and R2=2m, which is filled with a material of conductivityσ=1+2/R. (台科大電子研究所)
  37. The space between a parallel-plate capacitor of area S=1m2 is filled with a dielectric whose conductivity varies linearly fromσ=1×107(S/m) at one plate (y=0) toσ=2×107(S/m) at the other plate (y=d=1m). Neglecting fringing effect, find the resistance between the plates. (元智電研)
  38. A conducting material of uniform thickness h=0.5m and conductivityσ=2(S/m) has the shape of a quarter of a flat circular washer, with inner radius a=1m and outer radius b=2m. Determine the resistance between the end faces. (電信高員特考)
  39. A ground connection is made by burying a hemispherical conductor of radius 25 mm in the earth with its base up. Assuming the earth conductivity to beσS/m, find the resistance of the conductor to far-away points in the ground.
  40. Show that the mutual inductance , where L1 and L2are the self inductances of two coupled circuits.[清大核工所]
  41. Describe the corresponding physical parameters between the magnetic circuitsand the electric circuits.
  42. Assume thatN turns of wire are tightly wound on a toroidal frame of a rectangular cross section. Then, assuming the permeability of the medium to beμ0, find the self-inductance of the toroidal coil. [台大電研]
  43. Determine the inductance per unit length of an air coaxial transmission line that has a solid inner conductor of radius a and a very thin outer conductor of radius b. [台科大電機所]
  44. Determine the mutual inductance between a very long, straight wire and a conducting circular loop of radius b=2m. The distance between the wire and the center of the loop is 5m.[清大物理研究所]
  45. A current I flows in a long solenoid with n closely wound coil-turns per unit length. The cross-sectional area of its iron core, which has permeability μ, is S. Determine the force acting on the core if it is withdrawn to the position. [高考]
  46. Find the total magnetic flux through a circular 1000-turn toroid with a rectangular cross section of height h=1m and current I=1A. The inner and outer radii of the toroid are a=2m and b=3m, respectively. [台大電研]
  47. A current I=1A flows in a long solenoid with 1000 closely wound coil-turn per unit length. The cross-section area of its iron core, which has permeabilityμ=2000μ0, is S=10m2. Determine the force acting on the core if it is withdrawn.
  48. The following characteristics have been measured on a lossy transmission line at 100 MHz, Z0=50Ω, α=1.15×10-3 (Np/m), and β=0.8π(rad/m). Determine R, L, G, andC. [交大電子研究所類似題]
  49. Show that LC=μεand C/G=ε/σfor a transmission line.[台大電研類似題]
  50. Show that the attenuation constant is (Np/m), where PL(z) and P(z) are the time-average power loss and total input power, respectively.[台大電研類似題]
  51. Show that the input characteristic impedance of a transmission line connected to a load impedance ZL is Zi=, wherel, Z0and γ are the length, the characteristic impedance, and the propagation constant of the transmission line, respectively. [交大電子、清大電機研究所類似題]
  52. Show that a maximum power is transferred from a voltage source with an internal impedance Zg to a load impedance ZL over a lossless transmission line when Zi=Zg*, where Zi is the impedance looking into the loaded line. What is the maximum power transfer efficiency?
  53. A 2m lossless air-spaced transmission line having a characteristic impedance 50Ωis terminated with an impedance 40+j30 (Ω) at an operating frequency of 200 MHz. Find the input impedance.[台大電研類似題]
  54. It is desired to construct a 50Ωuniform transmission line using polyethylene (εr=2.25) as the dielectric medium. Assuming negligible losses, (a) find the distance of separation for a 300Ωtwo-wire line, where the radius of the conducting wires is 0.6mm; and (b) find the inner radius of the outer conductor for a 75Ωcoaxial line, where the radius of the center conductor is 0.6mm.[台大電研類似題]
  55. Consider a transmission line made of two parallel brass stripsσc=1.6×107(S/m) of width 20mm and separated by a lossy dielectric slab(μ=μ0,εr=3,σ=1×10-3) of thickness 2.5mm. The operating frequency is 500 MHz. Calculate R, L, G, C,Z0and γ.[交大光電研究所類似題]
  56. A generator with an open-circuit voltage and internal impedance 40+j30 (Ω) is connected to a 50Ωdistortionless line. The line has a resistance of 0.5 (Ω/m), and its lossy dielectric medium has a loss tangent of 0.18%. The line is 50m long and is terminated in a matched load. Find the instantaneous expressions for the voltage and current at an arbitrary location on the line.[台大電機期末考類似題]
  57. A transmission line of characteristic impedance R0=50Ωis to be matched to a load impedance ZL=40+j10 (Ω) through a length l’ of another transmission line of characteristic impedance R0’. Find the required l’ and R0’ for matching.[交大電信研究所]
  58. The standing-wave ratio on a lossless 300Ωtransmission line terminated in an unknown load impedance is 2.0, and the nearest voltage minimum is at a distance 0.3λfrom the load. Determine (a) the reflection coefficient Γof the load, (b) the unknown load impedance ZL, and (c) the equivalent length and terminating resistance of a line, such that the input impedance is equal to ZL.[中山電研]
  59. For an air-filled rectangular waveguide of dimension a=3.75mm and b=1.875mm. (a) Define the cutoff frequency and the dominant mode. (b) Find the frequency range which only allows the propagation of the dominant mode in this waveguide.[台大電信研究所]

(Sol.)∵a=2b,∴the dominant mode is TE10 mode and the next higher-order mode is TE01 mode.