Page 43 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 25

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1.25 Given point P(r = 0.8,θ = 30 ,φ = 45 ) and E = 1/r [cos φ a r +
(sin φ/ sin θ) a φ ], find (a) E at P;(b) |E| at P;(c)a unit vector in the
direction of E at P.
1.26 Express the uniform vector field F = 5a x in (a)cylindrical components;
(b) spherical components.
1.27 The surfaces r = 2 and 4, θ = 30 and 50 , and φ = 20 and 60 identify a
◦
◦
◦
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closed surface. Find (a) the enclosed volume; (b) the total area of the
enclosing surface; (c) the total length of the twelve edges of the surface;
(d) the length of the longest straight line that lies entirely within the surface.
1.28 State whether or not A = B and, if not, what conditions are imposed on A
and B when (a) A · a x = B · a x ;(b) A × a x = B × a x ;(c) A · a x = B · a x and
A × a x = B × a x ;(d) A · C = B · C and A × C = B × C where C is any
vector except C = 0.

1.29 Express the unit vector a x in spherical components at the point: (a) r = 2,
θ = 1 rad, φ = 0.8 rad; (b) x = 3, y = 2, z =−1; (c) ρ = 2.5,φ = 0.7 rad,
z = 1.5.
1.30 Consider a problem analogous to the varying wind velocities encountered by
transcontinental aircraft. We assume a constant altitude, a plane earth, a flight
along the x axis from 0 to 10 units, no vertical velocity component, and no
change in wind velocity with time. Assume a x to be directed to the east and
a y to the north. The wind velocity at the operating altitude is assumed to be:
2
(0.01x − 0.08x + 0.66)a x − (0.05x − 0.4)a y
v(x, y) =
1 + 0.5y 2
Determine the location and magnitude of (a) the maximum tailwind
encountered; (b) repeat for headwind; (c) repeat for crosswind; (d)Would
more favorable tailwinds be available at some other latitude? If so, where?