**Unformatted text preview: **ECE 329 Homework 3 Due: Tuesday September 17, 2019, 6pm 1. Is E = z xˆ − y yˆ + xˆ
z V/m a possible electrostatic eld in free space? Explain.
2. In free space, nd the electric eld E and static charge density ρ(x, y, z) if the scalar
−y
potential is V = sin(x)e z 2 V. Briey explain why E is a conservative eld.
3. Given that E = 1ˆ
x + sin(z)ˆ
y + y cos(z)ˆ
z V/m, determine the potential V (1, 2, 3) if
V (0, 0, 0) = 3V.
4. Given the two elds E = yˆ
x ± xˆ
y V/m,
¸ a) Determine the circulation C E · dl for a triangular path C traversing in order
its vertices at (x, y, z) = (1, −1, 0), (−1, −1, 0), and (1, 1, 0) m. Hint: dl ≡
dxˆ
x +dy yˆ+dz zˆ always. On the slant edge of C , x = y and z = 0 so dl = (ˆ
x + yˆ)dx
and E = xˆ
x ± xˆ
y.
˜
b) Verify your calculations by using Stokes' theorem to nd S (∇ × E)·dS. Hint:
Think of the area of the region bounded by C .
5. Between a pair of innitesimally thin sheets located on the z = −2 and z = 2 m
surfaces, a constant electric eld is observed to be E = −4ˆ
z V/m. What is the
voltage drop (or rise) Vp from the z = 2 m plane to the z = −2 m plane?
6. Consider a static sheet of charge ρ(x, y, z) = ρ0 δ(z) C/m3 .
a) Write down the expression for D as a vector above and below the plate.
b) Label z > 0 as region 1 and z < 0 as region 2 and aˆn = zˆ which points from
region 2 into region 1. Using your result from part (a), show that D1n − D2n ≡
a
ˆn · (D1 − D2 ) = ρ0 .
7. Consider a static charge distribution given by ρ0 = 3δ(y) + ρ00 δ(y − 2) C/m3 in free
space. Along the yˆ axis (i.e., for x = z = 0), the displacement eld is known to be
D = 2ˆ
y −2ˆ
z C/m2 for 0 < y < 2 m and Dy = 1 C/m2 for y > 2 m. Note that this eld
is a superposition of the eld generated by ρ0 together with a constant background
eld generated by a far away sheet charge source with surface charge density ρs .
a) Determine the unknown charge density ρ00 (be sure to include the units of your
answer). Hint: Use the boundary conditions from the previous problem.
b) Determine D(0, y, 0) for the region y > 2 m.
c) Determine D(0, y, 0) for the region y < 0 m.
d) Determine ρs and write two possible expressions for the volumetric charge density
ρ(x, y, z) of the far away sheet charge if its distance from the yˆ axis is known to
be 30 m. 1 8. Three important vector identities which are true for any scalar eld f (x, y, z) and
vector eld A(x, y, z) are:
∇ × (∇f ) = 0
∇ · (∇ × A) = 0
∇ × (∇ × A) = ∇(∇ · A) − ∇2 A where
2 ∂
∇2 A ≡ ( ∂x
2 + ∂2
∂y 2 + ∂2
)A
∂z 2 is the Laplacian of A and ∇(∇ · A) is the gradient of the divergence of A.
Verify the identities for f = x + yz , A = (x + z)yˆ
x + (x − z)ˆ
z by calculating each side of
each identity and showing them to be the same. There is no bonus question this week. 2 ...

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- Summer '08
- FRANKE
- Electrostatics, Electric charge, Fundamental physics concepts, charge density