Electric Field of Charged Rod (1)
y
x
• Charge per unit length: λ = Q/L
• Charge on slice dx: dq = λdx
dE
+ + + + + +++ + + + +
D
• Electric field generated by slice dx: dE =
dq = λ dx
x
L
kλdx
kdq
=
x2
x2
• Electric field generated by charged rod:
E = kλ
Z
D+L
D
»
»
–
–
1
dx
1
1 D+L
kQ
−
=
kλ
=
kλ
−
=
x2
x D
D
D+L
D(D + L)
kQ
D2
kλ
• Limiting case of very long rod (L ≫ D): E ≃
D
• Limiting case of very short rod (L ≪ D): E ≃
27/1/2009
[tsl31 – 5/13]
Electric Field of Charged Rod (2)
• Charge per unit length: λ = Q/L
• Charge on slice dxs : dq = λdxs
• Trigonometric relations:
yp = r sin θ,
−xs = r cos θ
yp dθ
xs = −yp cot θ, dxs =
sin2 θ
• dE =
kλdxs
kλdθ
kλdxs
2
=
sin
θ
=
r2
yp2
yp
θ2
kλ
kλ
• dEy = dE sin θ =
sin θdθ ⇒ Ey =
yp
yp
Z
kλ
kλ
• dEx = dE cos θ =
cos θdθ ⇒ Ex =
yp
yp
Z
sin θdθ = −
θ1
θ2
θ1
cos θdθ =
kλ
(cos θ2 − cos θ1 )
yp
kλ
(sin θ2 − sin θ1 )
yp
27/1/2009
[tsl32 – 6/13]
Electric Field of Charged Rod (3)
Symmetry dictates that the resulting electric field is directed radially.
• θ2 = π − θ1 ,
⇒ sin θ2 = sin θ1 ,
cos θ2 = − cos θ1 .
L/2
• cos θ1 = p
.
2
2
L /4 + R
• ER = −
• Ez =
kλ
kλ
L
p
.
(cos θ2 − cos θ1 ) =
R
R
L2 /4 + R2
kλ
(sin θ2 − sin θ1 ) = 0.
R
kQ
.
2
R
2kλ
• Small distances (R ≪ L): ER ≃
R
~ = 2kλ R̂.
• Rod of infinite length: E
R
• Large distance (R ≫ L): ER ≃
27/1/2009
[tsl33 – 7/13]
Electric Field of Charged Rod (4)
Symmetry dictates that the resulting electric field is directed radially.
• Charge per unit length: λ = Q/L
• Charge on slice dx: dq = λdx
• dE =
kλdx
kdq
=
r2
x2 + y 2
dEy
kλydx
• dEy = dE cos θ = p
= 2
(x + y 2 )3/2
x2 + y 2
"
#+L/2
Z +L/2
kλydx
kλyx
• Ey =
=
p
2
2 3/2
y 2 x2 + y 2
−L/2 (x + y )
−L/2
• Ey =
kλL
kQ
p
= p
y (L/2)2 + y 2
y (L/2)2 + y 2
• Large distance (y ≫ L): Ey ≃
kQ
y2
2kλ
• Small distances (y ≪ L): Ey ≃
y
y
dE
dq
x
+ + + + + + + + + ++ + ++ + + + + ++
−L/2
+L/2
27/1/2009
[tsl33 – 8/13]