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EE140 MT1 constant

创建日期：2024-04-11 17:02:55

所属分类：电气工程electric engineering

标签： EE140 MT1

constant

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EE140 MT1 Spring

1. A thin wire, carrying a positive and uniform linear charge density , is bent into a quarter-circle with a radius
R, as shown in the figure. (a) Find the electric field vector at a point (0, 0, z). (b) Find the electric potential at
the same location using = −∫ ⃗ ∙ ℓ⃗ .

x
z
y
R

1. A thin wire, carrying a positive and uniform linear charge density , is bent into a quarter-circle with a radius
R, as shown in the figure. (a) Find the electric field vector at a point (0, 0, z). (b) Find the electric potential at
the same location using = −∫ ⃗ ∙ ℓ⃗ .
⃗ ()
2
〈−, −, 〉

=

3
〈−, −, 〉
= √2 + 2 = √2 + 2 + 2 = constant
= −
2
3
∫

2
0
= −
2
3
=

=

3
∫

2
0
=

23

⃗ =

3
〈−,−,

2
〉

= − ∫⃗ ∙ = −

2
∫

[2 + 2]3/2

∞

∞

= 2 + 2, = 2
∫

[2 + 2]3/2

∞
=
1
2
∫−3/2 = −−
1
2 = −[
1
√2 + 2
]
∞

= −
1
√2 + 2

=

2√2 + 2

Check with direct integral:
=

=

, =

∫
/2
0

∙

2
=

2
=

2√2 + 2

r
x
z
dq (x,y,0)
dE
y
R

(0,0,z)
Why ∞ to z? Can it be -∞ to z?
Why dz? Can it be dx? dy?
by symmetry
2. The electric field in front of a thin charged circular disc at a distance z away
from the center is given by (from lecture, homework and quiz problems):
⃗ = ̂

2
[1 −

√2 + 2
].
Use this result to find the magnitude of the electric field at the tip of a right-
circular cone has a base radius R, height h, and a uniform charge density  as
shown in the figure. z
h
R
2. The electric field in front of a thin charged circular disc at a distance z away
from the center is given by (from lecture, homework and quiz problems):
⃗ = ̂

2
[1 −√2 + 2].
Use this result to find the magnitude of the electric field at the tip of a right-
circular cone has a base radius R, height h, and a uniform charge density  as
shown in the figure. z

2 ↔ 2, → , ℎ > > 0
() =
()
2
[1 −√2 + 2]=ℎ=2∫[1 −√2 + (ℎ )2]
ℎ0=2∫ [1 −ℎ√ℎ2 + 2]
ℎ0=ℎ2[1 −ℎ√ℎ2 + 2]
h
R
z
r
h
R

3. An infinitely long thin wire carrying a uniform charge density , is placed in front of the grounded conductor
at (d, s, 0) and parallel to the z-axis as shown. (a) Find the electric field at any point (x, y, z) in the first quadrant.
You do not need to simplify your answer, as long as all the variables are CLEARLY defined. (b) Find the
surface charge density induced on the xz-plane (i.e., at y = 0).
Grounded
conductor
line
charge s
d
y
x
air
3. An infinitely long thin wire carrying a uniform charge density , is placed in front of the grounded conductor
at (s, d, 0) and parallel to the z-axis as shown. (a) Find the electric field at any point (x, y, z) in the first quadrant.
You do not need to simplify your answer, as long as all the variables are CLEARLY defined. (b) Find the
surface charge density induced on the xz-plane (i.e., at y = 0).

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