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Problem 5.23 Find the magnetic vector potential of a finite segment of straight wire carrying a current I. [Put the wire on the z axis, from z1 to z2, and use Eq. 5.66.]Check that your answer is consistent with Eq. 5.37.

A standard X-band (8.2-12.4 GHz) rectangular waveguide with inner dimensions of 0.9 in.(2.286 cm) by 0.4 in. (1.016 cm) is filled with lossless polystyrene (ɛr = 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) Waye impedance. (d) Phase velocity (in m/s). (e) Group velocity (in m/s).

3.56 Determine if each of the following vector fields is solenoidal, conservative, or both: \mathbf{A}=\hat{\mathbf{x}} x^{2}-\hat{\mathbf{y}} y 2 x y \mathbf{B}=\hat{\mathbf{x}} x^{2}-\hat{\mathbf{y}} y^{2}+\hat{\mathbf{z}} 2 z \mathbf{C}=\hat{\mathbf{r}}(\sin \phi) / r^{2}+\hat{\phi}(\cos \phi) / r^{2} \mathbf{D}=\hat{\mathbf{R}} / R \mathbf{E}=\hat{\mathbf{r}}\left(3-\frac{F}{1+P}\right)+\mathbf{z} z \mathbf{G}=\hat{\mathbf{x}}\left(x^{2}+z^{2}\right)-\hat{\mathbf{y}}\left(y^{2}+x^{2}\right)-\hat{\mathbf{z}}\left(y^{2}+z^{2}\right) \mathbf{H}=\hat{\mathbf{R}}\left(R e^{-R}\right)

Problem 4.19 Suppose you have enough linear dielectric material, of dielectric constant e,, to half-fill a parallel-plate capacitor (Fig. 4.25). By what fraction is the capacitance increased when you distribute the material as in Fig. 4.25(a)? Howa bout Fig. 4.25(b)? For a given potential difference V between the plates, find E,D, and P, in each region, and the free and bound charge on all surfaces, for both

Problem 6.12 An infinitely long cylinder, of radius R, carries a "frozen-in" magnetization, parallel to the axis, M = ks î, where k is a constant and s is the distance from the axis; there is no free current anywhere. Find the magnetic field inside and outside the cylinder by two different methods: (a) As in Sect. 6.2, locate all the bound currents, and calculate the field they produce. (b) Use Ampère's law (in the form of Eq. 6.20) to find H, and then get B from Eq. 6.18. (Notice that the second method is much faster, and avoids any explicit reference to the bound currents.)

. A half-wave dipole is radiating into free-space. The coordinate system is defined so that the origin is at the center of the dipole and the z-axis is aligned with the dipole. Input power to the dipole is 100 W. Assuming an overall efficiency of 50%, find the power density (in W/m2) atr = 500 m, 0 = 60°, = 0°.

4-3.The primary of a transformer has 200 turns and is excited by a 60-Hz, 220-V source. a. Find the maximum value of the core flux. b. Let a voltage v = 155.5 sin 377t + 15.5 sin 1131t (V) be applied to the transformer primary. Neglecting leakage, determine the instantaneous and rms values of the core flux.

Problem 2.25 Using Eqs. 2.27 and 2.30, find the potential at a distance z above the center of the charge distributions in Fig. 2.34. In each case, compute E =-V, and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectively. Suppose that we changed the right-hand charge in Fig. 2.34a to -q; what then is the potential at P? What field does that suggest? Compare your answer to Prob. 2.2, and explain carefully any discrepancy.

7.42 A team of scientists is designing a radar as a probe for measuring the depth of the ice layer over the antarctic land mass. In order to measure a detectable echo due to the reflection by the ice-rock boundary, the thickness of the ice sheet should not exceed three skin depths. If & = 3 andE" = 10-2 for ice and if the maximum anticipated ice thickness in the area under exploration is 1.2 km, what frequency range is use able with the radar?

4.61 With reference to Fig. P4.61, charge Q is located at a distance d above a grounded half-plane located in the x-y plane and at a distance d from another grounded half-plane in the x-z plane. Use the image method to (a) Establish the magnitudes, polarities, and locations of the images of charge Q with respect to each of the two ground planes (as if each is infinite in extent). (b) Find the electric potential and electric field at an arbitrary point P = (0,y,z).