高等电磁理论复习题

1. Derive the differential form of the continuity equation from Maxwell’s equations.
2. Derive the integral forms of Maxwell’s equations and the continuity equations, from the corresponding ones in differential form
3. The electric flux density inside a cube is given by (a) ()ˆ3x x =+D a
, (b) ()2ˆ4y y =+D a
. Find the total electric charge enclosed inside the cubical volume when the cube is in the first octant with three edges coincident with the x , y , z axes and one corner at the origin. Each side of the cube is 1m.
4. The electric field inside a circular cylinder of radius a and
height h is given by
()220ˆ36z c b z h h ε⎡⎤=-+-⎢⎥⎣⎦
E a where c and b are constants. Assuming the medium within the
cylinder is free space, find the total charge enclosed within the
cylinder.
5. The instantaneous electric field inside a source-free, homogeneous, isotropic, and linear medium is given by
()()()ˆˆcos x y A x y B x y t ω⎡⎤=++-⎣⎦E a a
Determine the relations between A and B .
6. The magnetic flux density produced on its plane by a current-carrying circular loop of radius a = 0.1 m , placed on the xy plane at z = 0, is given by
()1210ˆcos 1500125z t πρ
-=+B a Wb / m 2 where ρ is the radial distance in cylindrical coordinates.
(a) Find the total flux in the z direction passing through the loop.
(b) Find the electric field at any point ρ within the loop.
7. The instantaneous magnetic flux density in free space is given by
()()()()ˆˆcos 2sin cos 2cos x x y y B y t z B y t z ωπωπ=-+-B a
a where Bx and By are constants. Assuming there are no sources, determine the electric displacement current density.
8. The displacement current density within a source-free cube centered about the origin is given by
2ˆˆˆd x y z yz y xyz =++i a
a a Each side of the cube is 1 m and the medium within it is free space. Find the displacement current leaving, in the outward direction, through the surface of the cube.
9. The time-harmonic complex electric field radiated in free space by a linear radiating element is given by
ˆˆr r E E θθ=+E a
a 0j 020cos 11j r r E E e r
r βθβ-⎡⎤=+⎢⎥⎣⎦ ()0j 00200sin 11j 12j r E E e r r r βθβθββ-⎡⎤=+-⎢⎥⎢⎥⎣⎦
where ˆr a
and ˆθa are unit vectors in the spherical direction r and θ, E 0 is a constant,
and 0β=. Determine the corresponding spherical magnetic field
components.
10. A time-harmonic electromagnetic field traveling in free space and perpendicularly incident upon a flat surface of distilled water (ε = 81ε0, μ = μ0), as shown in following figure, creates a reflected field on the free-space side of the interface and a transmitted field on the water side of the interface. Assuming the
incident (E i ), reflected (E r ), and transmitted (E t )
electric fields are given, respectively, by
0j 0ˆz i x E e β-=E a
0+j 00ˆz r x E e βΓ=E a
j 00ˆt z x T E e β-=E a
determine the coefficients Γ0 and T 0. E 0 is a constant, 0β=β=
11. The current density through a cylindrical wire of square cross section as shown in the following figure is given by
()()2100ˆa x a y z J e ⎡⎤--+-⎣⎦=J a
where J 0 is a constant. Assuming that each side of the wire is 2 × 10-2
m, find the current flow through the cross section of the wire.
12. A 10-A current is pushed through a circular cross section of wire
of infinite length as shown in the following figure. Assuming that the
current density over the cross section of the wire decays from its
surface toward its center as
()4100ˆa z J e ρ--=J a A / m 2
where J 0 is the current density at the surface and the wire radius is a =
10-2 m, determine
(a) The current density at the surface of the wire.
(b) The depth from the surface of the wire through which the current
density has decayed to 36.8 percent of its value at the surface.
13. A uniform plane wave propagating in a medium with relative permittivity of 4 is incident normally upon a dielectric medium with dielectric constant of 9. Assuming both media are nonferromagnetic and lossless, determine (a) the reflection and transmission coefficients and (b) the percentage of indicent power density that is reflected and transmitted.
14. A time-harmonic electromagnetic wave traveling in free
space is incident normally upon a perfect conducting planar
surface, as shown in the following figure. Assuming the
incident electric field is given by
0j 0ˆz i x E e β-=E a
find (a) the reflected electric field, (b) the incident and
reflected magnetic fields, and (c) the current density J s induced on the conducting surface.
15. A linearly polarized wave is incident on an isosceles right
triangle (prism) of glass, and it exits as shown in the following
figure. Assuming that the dielectric constant of the prism is 2.25,
find the ratio of the exited average power density S e to that of the
incident S i.
16. A parallel polarized uniform plane wave is incident
obliquely on a lossless dielectric slab that is embedded in
a free-space medium, as shown in the following figure.
Derive expressions for the total reflection and
transmission coefficients in terms of the electrical
constitutive parameters, thickness of the slab, and angle of
incidence.
17. A uniform plane wave at frequency of 104 Hz is traveling in air, and it is incident normally on a large body of salt water with constants of σ = 3 S / m and εr = 81. If the magnitude of the electric field on the salt water side of the interface is 10-3 V / m, find the depth (in meters) inside the salt water at which the magnitude of hte electric field has been reduced to 0.368 × 10-3 V / m.
18. A standard X-band (8.2 - 12.4 GHz) rectangular waveguide with inner dimensions of 0.9 in. (2.286 cm) and 0.4 in. (1.016 cm) is filled with lossless polystyrene (εr = 2.56). For the lowest-order mode of the waveguide, determine at 10 GHz the following values.
(a) Cutoff frequency (in GHz).
(b) Guide wavelength (in cm).
(c) Wave impedance.
(d) Phase velocity (in m / s).
(e) Group velocity (in m / s).。