Page 90 - Engineering Electromagnetics, 8th Edition

72 ENGINEERING ELECTROMAGNETICS

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3.9 A uniform volume charge density of 80 µC/m is present throughout the
region 8 mm < r < 10 mm. Let ρ ν = 0 for 0 < r < 8 mm. (a) Find the total
charge inside the spherical surface r = 10 mm. (b) Find D r at r = 10 mm.
(c)If there is no charge for r > 10 mm, find D r at r = 20 mm.
3.10 An infinitely long cylindrical dielectric of radius b contains charge within its
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volume of density ρ v = aρ , where a is a constant. Find the electric field
strength, E, both inside and outside the cylinder.
3.11 In cylindrical coordinates, let ρ ν = 0 for ρ< 1 mm, ρ ν = 2 sin(2000
3
πρ) nC/m for1mm <ρ < 1.5 mm, and ρ ν = 0 for ρ> 1.5 mm. Find D
everywhere.
3.12 The sun radiates a total power of about 3.86 × 10 26 watts (W). If we imagine
the sun’s surface to be marked off in latitude and longitude and assume
uniform radiation, (a) what power is radiated by the region lying between
latitude 50 N and 60 N and longitude 12 W and 27 W? (b) What is the
◦
◦
◦
◦
2
power density on a spherical surface 93,000,000 miles from the sun in W/m ?
3.13 Spherical surfaces at r = 2, 4, and 6 m carry uniform surface charge
2
2
densities of 20 nC/m , −4 nC/m , and ρ S0 , respectively. (a) Find D at r = 1,
3, and5m.(b) Determine ρ S0 such that D = 0at r = 7m.
3.14 A certain light-emitting diode (LED) is centered at the origin with its surface
in the xy plane. At far distances, the LED appears as a point, but the glowing
surface geometry produces a far-field radiation pattern that follows a raised
2
cosine law: that is, the optical power (flux) density in watts/m is given in
spherical coordinates by
2
cos θ 2
P d = P 0 a r watts/m
2πr 2
where θ is the angle measured with respect to the direction that is normal to
the LED surface (in this case, the z axis), and r is the radial distance from the
origin at which the power is detected. (a)In terms of P 0 , find the total power
in watts emitted in the upper half-space by the LED; (b) Find the cone angle,
θ 1 , within which half the total power is radiated, that is, within the range
0 <θ <θ 1 ;(c)An optical detector, having a 1-mm cross-sectional area, is
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positioned at r = 1m and at θ = 45 , such that it faces the LED. If one
◦
milliwatt is measured by the detector, what (to a very good estimate) is the
value of P 0 ?
3.15 Volume charge density is located as follows: ρ ν = 0 for ρ< 1mm and for
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ρ> 2 mm, ρ ν = 4ρ µC/m for 1 <ρ < 2 mm. (a) Calculate the total charge
in the region 0 <ρ <ρ 1 ,0 < z < L, where 1 <ρ 1 < 2 mm. (b) Use
Gauss’s law to determine D ρ at ρ = ρ 1 . (c)Evaluate D ρ at ρ = 0.8 mm,
1.6 mm, and 2.4 mm.
3.16 An electric flux density is given by D = D 0 a ρ , where D 0 is a given constant.
(a) What charge density generates this field? (b)For the specified field, what