Electrostatics:

1. A circular disk of radius 𝑏 is uniformly charged with ρS C/m2. If the disk lies on the x = 0 plane
with its axis along the x-axis,
(a) Show that at (ℎ, 0, 0)
𝜌𝑆 ℎ
𝑬 = {1 − 2 }𝒂
𝜖0 [ℎ + 𝑏 2 ]1/2 𝒙
(b) From this, derive the E field due to an infinite sheet of charge on the x = 0 plane.
2. A point charge 200 pC is located at (4, 1, -3) while the x-axis carries charge 2 nC/m. If the plane
z = 5 also carries charge 10 nC/m2, find E at (1, 1, 1).
3. In free space, 𝑉 = 𝑥 2 𝑦(𝑧 + 5) V. Find
(a) E at (3, 4, -6)
(b) the charge within the cube 0 < x, y, z < 2.
4. The electric field intensity in polystyrene (𝜖𝑟 = 5) filling the space between the plates of a
parallel-plate capacitor is 20 kV/m. The distance between the plates is 1.5 mm. Calculate:
(a) D (electric flux density)
(b) P (polarization)
(c) The surface charge density of free charge on the plates (ρs)
(d) The surface density of polarization charge (ρps)
(e) The potential difference between the plates.
5. Given that 𝑬𝟏 = 5𝒂𝒙 − 𝟑𝒂𝒚 + 10𝒂𝒛 , V/m in below figure, find: (a) P1 (the polarization
vector in medium 1) (b) E2 and the angle E2 makes with the y-axis, (c) the energy density in
each region.

6. Two homogeneous dielectric regions 1 (ρ < 2 cm) and 2 (ρ > 2 cm) have dielectric constants
3.5 and 1.5, respectively. If 𝑫𝟐 = 10𝒂𝝆 − 6𝒂𝝋 + 5𝒂𝒛 nC/m2, calculate: (a) E1 and D1, (b) P2
and ρpv2, (c) the energy density for each region.
7. Two lossy homogeneous dielectric media with dielectric constants 𝜖𝑟1 = 3 and 𝜖𝑟2 = 5 and
conductivities 𝜎1 = 15 mS and 𝜎2 = 12 mS are in contact at z = 0 plane. In the 𝑧 > 0 region
(medium 1) a uniform electric field 𝑬𝟏 = 10𝒂𝒙 − 40𝒂𝒛 (V/m) exists. Find (a) E2 in medium 2,
(b) J1 and J2, (c) the angles that J1 and J2 make with the z = 0 plane, and (d) the surface charge
density at the surface.
8. To verify that 𝑬 = 𝑦𝑧𝒂𝒙 + 𝑥𝑧𝒂𝒚 + 𝑥𝑦𝒂𝒛 V/m is truely an electric field, show that
(a) 𝛁 × 𝑬 = 0
(b) ∮𝐿 𝐄. 𝑑𝐥 = 0, where L is the edge of the square defined by 0 < 𝑥, 𝑦 < 4, 𝑧 = 3.
9. A point charge of 60 nC is located at the origin while plane y = 5 carries charge 20nC/m2. Find
D at (0,4, 3).
10. Determine the total charge
(a) On the line 0 < x < 5 m if 𝜌𝐿 = 15𝑥 2 mC/m
(b) On the cylinder ρ = 5, 0 < z < 4 m if 𝜌𝑆 = 𝜌𝑧 2 nC/m2
20
(c) Within the sphere 𝑟 = 4 m, if 𝜌𝑣 = C/m3
𝑟 sin 𝜃
11. Three-point charges -2 nC, 5 nC, and 6 nC are located at (0, 0, 0), (0, 0, 1), and (1, 0, 0),
respectively. Find the energy in the system.
12. Let 𝑬 = 𝑥𝑦𝒂𝒙 + 2𝑥 2 𝒂𝒚 and assuming a free space medium, find
(a) Electric flux density D.
(b) The volume charge density 𝜌𝑣 .
13. For the current density 𝑱 = 20𝑧 𝑠𝑖𝑛2 𝜑 𝒂𝝆 A/m2, find the current through the cylindrical
surface ρ = 2, 1 ≤ 𝑧 ≤ 2 m.
14. Determine the relaxation time for Distilled water (𝜎 = 10−4 S/m, 𝜖 = 80𝜖0 ).
15. Three 2 µC point charges are located in air at the corners of an equilateral triangle that is 10
(cm) on each side. Find the magnitude and direction of the force experienced by each charge.
16. Assume that the z = 0 plane separates two lossless dielectric regions with 𝜖𝑟1 = 3 and 𝜖𝑟2 =
5. If we know that E1 in region 1 is 3𝑦𝒂𝒙 − 4𝑥𝒂𝒚 + (7 + 𝑧)𝒂𝒛 . Find E2 and D2 at the interface
in region 2.
17. Determine the electric field due to 𝑉 = 𝜌2 (𝑧 + 2) sin 𝜑.
18. Determine the work necessary to transfer charges 𝑄1 = 2 mC and 𝑄2 = −3 mC from infinity
to points (-2, 6, 2) and (4, -4, 0), respectively.
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