ELE401 - Field Theory: Tutorial Problems
Week 3
~ =
1. From first principles using dE
dQ(~
r −~
r0 )
4πεo |~
r −~
r 0 |3 .
~
(a) Develop the E-field
expression for the quarter circle of line charge
~
density as shown on the figure 1. The E-field
to be found at point
P (0, 0, z).
z
P(0,0,z)
z'
a
ρ L[C/m]
y
x
Figure 1: Quarter circle of line charge density
(b) Solve 1(a) for the case where the line charge forms a complete circle
(i.e. φ : 0 → 2π).
(c) Drop the circle of line charge down onto the x-y plane by letting
z 0 → 0.
2. (a) Use the expression derived in 1(c) to develop the incremental expression for a narrow circular strip of surface charge ρs as shown on the
figure 2.
~ to solve for a disk or washer
Then use the new expression for dE
shape surface of charge such that a ≤ ρ ≤ b.
(b) Let a → 0, then . . .
1
z
P(0,0,z)
ρ s [C/m ]
2
b
a
y
ρ'
dρ'
x
Figure 2: A narrow circular strip of surface charge ρs .
(c) Finally, let b → ∞, then the entire x-y plane becomes a sheet of
surface charge.
3. A sphere with a hollow core is centered at the origin, as shown on figure 3.
The hallow part has a radius of a; the other radius of the sphere is b [m].
The material part of the sphere carries a volume charge density ρv =
αr [c/m3 ]. Please note ρv = ρv (r).
z
ρ v[C/m ]
3
b
a
ρ
0
Cross
section view
Figure 3: Cross section of a sphere with a hollow core.
(a) Solve for the total charge contained in the sphere.
2
(b) Using a Gauss’ law and symmetry solve for the electric flux density
~ in the following regions.
vector field D
i. r < a
ii. a ≤ r ≤ b
iii. r > b
~ ·D
~ in all three regions.
(c) Solve for ∇
~ is given by:
4. Given that the electric flux density D
~ = ρ sin φaˆρ + ρ2 z aˆφ + z cos φaˆz [c/m2 ]
D
(1)
(a) Solve for the total flux crossing the closed surface shown on the figure 4.
I
~ · d~s [C]
(2)
ψ= D
s
z
z=b
y=a
y
x=a
ρ=a
x
Figure 4:
(b) Solve for the volume charge density ρv ,
~ ·D
~ [c/m3 ]
ρv = ∇
(c) Solve for the total charge Q enclosed by the surface.
Z
Q = ρv dv [C]
(3)
(4)
v
~ field is changed to:
5. Show that the divergence theorem still works if the D
~ = sin φ aˆρ + ρ2 z aˆφ + z cos φaˆz [c/m2 ]
D
ρ
3
(5)