1300 (W/m 2 ) (V/cm) = 275 (V/m) (A/cm) = (A/m). E = 990 (V/m), H = 2.63 (A/m).

Transcription

1 Homework #4 P8-16 There is a continuing discuss on radiation hazards to human health The following calculations will provide a rough comparison a) The US standard for personal safet in a microwe environment is that the power densit be less than 1 (mw/cm ) Calculate the corresponding standard in terms of electric field intensit In terms of magnetic field intensit b) It is estimated that the earth receives radiant energ from the sun at a rate of about 13 (kw/m ) on a sunn da Assuming a monochromatic plane we (which it is not), calculate the equivalent amplitudes of the electric and magnetic field intensit vectors P 1 (W/cm ) a) 75 (V/cm) = 75 (V/m) H (A/cm) = 78 (A/m) b) P 13 (W/m ) = 99 (V/m), H = 63 (A/m) P8-18 Assuming that the radiation electric field intensit of an antenna sstem is = a + a, find the epression for the erage outward power flow per unit area Since -field is epressed as above, the magnetic field can be found as 1 1 H aˆ ˆ ˆ R a a 1 1 P ˆ e H ar P8-19 From the point of view of electromagnetics, the power transmitted b a lossless coaial cable can be considered in terms of the Ponting vector inside the dielectric medium between the inner conductor and the outer sheath Assuming that a d-c voltage V applied between the inner conductor (of radius a) and the outer sheath (of inner radius b) causes a current I to flow to a load resistance, verif that the integration of the Ponting vector over the cross-sectional area of the dielectric medium equals the power VI that is transmitted to the load

2 From Gauss s law, aˆ r, where is the line charge densit on the inner conductor r a b V V ˆ dr ln ar b a rln( b/ a) I From Ampere s circuital law, H aˆ r VI Ponting vector: P H aˆ z r ln( b/ a) Power transmitted over cross-sectional area: VI b 1 P P ds rdrd V I S ln( b/ a) a r P8- A uniform sinusoidal plane we in air with the following phasor epression for electric intensit i(, z) = a1e j(6 + 8z) (V/m) is incident on a perfectl conducting plane at z = a) Find the frequenc and welength of the we b) Write the instantaneous epression for i(, z; t) and Hi(, z; t), using a cosine reference c) Determine the angle of incidence d) Find r(, z) and Hr(, z) of the reflected we e) Find 1(, z) and H1(, z) of the total field

3 P 8-6 Determine the condition under which the magnitude of the reflection coefficient equals that of the transmission coefficient for a uniform plane at normal incidence on an interface between two lossless dielectric media What is the standing-we ratio in db under this condition? For normal incidence: 1 + =, where 1 If = : < and 1 = 1 = 3 = 5 1 Thus, S 3 SdB = log13 = 954 (db) 1 P8-8 A uniform plane we in air with i(z) = a ep( jz) impinges normall onto the surface at z = of a highl conducting medium hing constitutive parameters,, and (/ >> 1) a) Find the reflection coefficient b) Derive the epression for the fraction of the incident power absorbed b the conducting medium c) Obtain the fraction of the power absorbed at 1 (MHz) if the medium is iron P9- The electric and magnetic fields of a general TM we treling in the +z-direction along a transmission line ma he both - and -components, and both components ma be functions of the transverse dimensions a) Find the relations among (, ), (, ), H(, ) and H(, ) b) Verif that all the four field components in part (a) satisf the two-dimensional Laplace s equation for static fields a) ( a ˆ a ˆ ) j( ah ˆ ah ˆ )

4 H, (1) H, () (3) ( ah ˆ ah ˆ ) j ( a ˆ a ˆ ) H, (4) H, (5) H H (6) From (1) and (5): (7) From () or (4): (8) H From (1) or (5): (9) H b) From (3): From (4), (5) and (6): (1) Combining (1) and (11), we he: Similarl, we can obtain:, H H (11), and H H P9-4 Consider a transmission line made of two parallel brass strips c = (S/m) of width (mm) and separated b a loss dielectric slab =, r = 3, = 1 3 (S/m) of thickness 5 (mm) The operating frequenc is 5 MHz a) Calculate the R, L, G, and C per unit length b) Compare the magnitudes of the aial and transverse components of the electric field c) Find and Z

5 z r 71 c c 5 P9-7 In the derivation of the approimate formulas of and Z for low-loss lines in Subsection 9-31, all terms containing the second and higher powers of (R/L) and (G/C) were neglected in comparison with unit At lower frequencies, better approimations than those given in qs (9-54) and (9-58) ma be required Find new formulas for and Z for low-loss lines that retain terms containing (R/L) and (G/C) Obtain the corresponding epression for phase velocit