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Find the gradient of the following scalar functions: Find the gradient of the following scalar functions: (a) T = 3 / (x2 + z2), (b) V = xy2 z4|, (c) U =z cos?????1 + r2), (d) W = e-R sin?, (e) S = 4x2e-z + y3, (f) N = 2 cos2 ?, (g) M = […]

Find the distance between the following pairs of points Find the distance between the following pairs of points: (a) P1 (1, 2, 3) and P2 (- 2, – 3, -2) in Cartesian coordinates, (b) P3 (1, ????????and P4 (3, ????????in cylindrical coordinates, (c) P5 (4, ??2, 0) and P6 (3, ??????in spherical coordinates. Is this […]

The three regions shown in Fig. 8-32 (P8.9) contain perfect The three regions shown in Fig. 8-32 (P8.9) contain perfect dielectrics. For a wave in medium 1 incident normally upon the boundary at z = – d, what combination of ?r2 and d produce no reflection? Express your answers in terms of ?r1, ?r3 and […]

Find and plot the normalized array factor and determine the Find and plot the normalized array factor and determine the half power beam width for a five-element linear array excited with equal phase and a uniform amplitude distribution. The interelement spacing is 34. Is this the question you were looking for? If so, place your […]

Find an expression for the unit vector Find an expression for the unit vector directed toward the point P located on the z-axis at a height h above the x–y plane from an arbitrary point Q(x, y, – 3) in the plane z = – 3. Is this the question you were looking for? If […]

The vertical displacement of a string is given by the harmonic The vertical displacement of a string is given by the harmonic function: y(x, t) = 6cos (16pt – 20px) (m), where x is the horizontal distance along the string in meters. Suppose a tiny particle were to be attached to the string at x […]

Figure P8.21 depicts a beaker containing a block of glass Figure P8.21 depicts a beaker containing a block of glass on the bottom and water over it. The glass block contains a small air bubble at an unknown depth below the water surface. When viewed from above at an angle of 60_, the air bubble […]

Figure 4-34a (P4.52 (a)) depicts a capacitor consisting of Figure 4-34a (P4.52 (a)) depicts a capacitor consisting of two parallel, conducting plates separated by a distance d. The space between the plates contains two adjacent dielectrics, One with permittivity ?1 and surface area A1 and another with ?2 and A2 the objective of this problem […]

A large plate is fabricated from a steel alloy that A large plate is fabricated from a steel alloy that has a plane strain fracture toughness of 55MPavm (50ksivin) If, during service use, the plate is exposed to a tensile stress of 200MPa (29,000psi), determine the minimum length of a surface crack that will lead […]

With reference to Fig. 4-37 (P4.56), charge Q is located With reference to Fig. 4-37 (P4.56), 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, […]