Page 123 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 105

CHAPTER 4 PROBLEMS
4.1 The value of E at P(ρ = 2, φ = 40 , z = 3) is given as E = 100a ρ
◦
− 200a φ + 300a z V/m. Determine the incremental work required to move a
20 µC charge a distance of 6 µm: (a)in the direction of a ρ ;(b)inthe
direction of a φ ;(c)in the direction of a z ;(d)in the direction of E;(e)inthe
direction of G = 2a x − 3a y + 4a z .
4.2 A positive point charge of magnitude q 1 lies at the origin. Derive an
expression for the incremental work done in moving a second point charge q 2
through a distance dx from the starting position (x, y, z), in the direction
of −a x .
4.3 If E = 120a ρ V/m, find the incremental amount of work done in moving
a 50-µC charge a distance of 2 mm from (a) P(1, 2, 3) toward Q(2, 1, 4); (b)
Q(2, 1, 4) toward P(1, 2, 3).
4.4 An electric field in free space is given by E = xa x + ya y + za z V/m. Find
the work done in moving a 1-µC charge through this field (a) from (1, 1, 1)
to (0, 0, 0); (b) from (ρ = 2, φ = 0) to (ρ = 2, φ = 90 ); (c) from (r = 10,
◦
θ = θ 0 )to(r = 10, θ = θ 0 + 180 ).
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4.5 Compute the value of A P G · dL for G = 2ya x with A(1, −1, 2) and
P(2, 1, 2) using the path (a) straight-line segments A(1, −1, 2) to B(1, 1, 2)
to P(2, 1, 2); (b) straight-line segments A(1, −1, 2) to C(2, −1, 2) to
P(2, 1, 2).
4.6 An electric field in free space is given as E = x ˆa x + 4z ˆa y + 4y ˆa z .Given
V (1, 1, 1) = 10 V, determine V (3, 3, 3).
2
4.7 Let G = 3xy a x + 2za y Given an initial point P(2, 1, 1) and a final point
Q(4, 3, 1), find G · dL using the path (a) straight line: y = x − 1,
2
z = 1; (b) parabola: 6y = x + 2, z = 1.
4.8 Given E =−xa x + ya y ,(a) find the work involved in moving a unit positive
charge on a circular arc, the circle centered at the origin, from x = a to
√
x = y = a/ 2; (b)verify that the work done in moving the charge around
the full circle from x = a is zero.
2
4.9 A uniform surface charge density of 20 nC/m is present on the spherical
surface r = 0.6cmin free space. (a) Find the absolute potential at
P(r = 1 cm, θ = 25 , φ = 50 ). (b) Find V AB ,given points A(r = 2 cm,
◦
◦
θ = 30 , φ = 60 ) and B(r = 3 cm, θ = 45 ,φ = 90 ).
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◦
◦
◦
2
4.10 A sphere of radius a carries a surface charge density of ρ s0 C/m .(a) Find
the absolute potential at the sphere surface. (b)A grounded conducting shell
of radius b where b > a is now positioned around the charged sphere. What
is the potential at the inner sphere surface in this case?
2
4.11 Let a uniform surface charge density of 5 nC/m be present at the z = 0
plane, a uniform line charge density of 8 nC/m be located at x = 0, z = 4,