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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
September 20, 2005
Lecture 4: The Scalar Electric Potential and the Coulomb Superposition
Integral
I. Quasistatics
Electroquasistatics (EQS)
Magnetquasistatics (MQS)
0
∇×E = −
∂
µ0 H ≈ 0
∂t
(
)
∇×E = −
∂
µ0 H
∂t
(
)
0
∂
ε0 E
∇×H = J+
∂t
( )
( )
∇ i ε0 E = ρ
∇×H = J+
∇iJ+
∂
∂t
( ε E)
(
)
∇ i µ0H = 0
0
∂ρ
=0
∂t
∇iJ = 0
( )
∇ i ε0 E = ρ
II. Irrotational EQS Electric Field
1. Conservative Electric Field
0
d
∫ E i ds = − dt ∫ µ
C
0
H i da ≈ 0
S
6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 4
Page 1 of 6
∫
b
E i ds =
C
a
b
∫ E i ds +
∫
I
II
a
E i ds = 0 ⇒
b
∫ E i ds
a
I
b
=
∫ E i ds
a
II
Electromotive Force
(EMF)
EMF between 2 points (a, b) independent of path
E field is conservative
()
r ref
( ) ∫ E i ds
Φ r
− Φ r ref =
r
Scalar
electric potential
b
r ref
b
a
a
r ref
∫ E i ds =
∫ E i ds + ∫ E i ds = Φ ( a) − Φ (r
ref
) + Φ ( r ) − Φ (b ) = Φ ( a ) − Φ ( b )
ref
2. The Electric Scalar Potential
_
_
_
r = x ix + y iy + z iz
_
_
_
∆r = ∆x i x + ∆y i y + ∆z i z
∆n = ∆r cos θ
6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 4
Page 2 of 6
∆Φ = Φ ( x + ∆x, y + ∆y, z + ∆z ) − Φ ( x, y, z )
= Φ ( x, y, z ) +
=
∂Φ
∂Φ
∂Φ
∆x +
∆y +
∆z − Φ ( x, y, z )
∂x
∂y
∂z
∂Φ
∂Φ
∂Φ
∆x +
∆y +
∆z
∂x
∂y
∂z
⎡ ∂Φ _
∂Φ _
∂Φ _ ⎤
=⎢
ix +
iy +
i i ∆r
∂y
∂z z ⎥⎦
⎣ ∂x
grad Φ = ∇Φ
_
∇ = ix
_
_
∂
∂
∂
+ iy
+ iz
∂x
∂y
∂z
_
grad Φ = ∇Φ = i x
r +∆r
∫
()
∂Φ _ ∂Φ _ ∂Φ
+ iy
+ iz
∂x
∂y
∂z
(
)
E i ds = Φ r − Φ r + ∆r = −∆Φ = −∇Φ i ∆r = E i ∆r
r
E = −∇Φ
∆Φ =
∆Φ
∆Φ
n i ∆r = ∇Φ i ∆r
∆r cos θ =
∆n
∆n
∇Φ =
∆Φ
∂Φ
n=
n
∆n
∂n
The gradient is in the direction perpendicular to the equipotential
surfaces.
III. Vector Identity
∇×E = 0
E = −∇Φ
∇ × ( ∇Φ ) = 0
6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 4
Page 3 of 6
IV. Sample Problem
Φ ( x, y ) =
V0 xy
a2
(Equipotential lines hyperbolas: xy=constant)
⎡ ∂Φ _
∂Φ _ ⎤
E = −∇Φ = − ⎢
ix+
i ⎥
∂y y ⎦
⎣ ∂x
=
_ ⎞
−V0 ⎛ _
y ix+ x iy ⎟
2 ⎜
a ⎝
⎠
Electric Field Lines [lines tangent to electric field]
dy Ey
x
=
= ⇒ ydy = xdx
dx Ex
y
y2
x2
=
+C
2
2
y2 − x2 = y20 − x20
[lines pass through point ( x0 , y0 ) ]
(hyperbolas orthogonal to xy)
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 4
Page 4 of 6
V. Poisson’s Equation
∇ i E = ∇ i ( −∇Φ ) = ρ
ε0 ⇒ ∇ 2 Φ = − ρ ε 0
_
_
⎡_ ∂
∂
∂ ⎤ ⎡ ∂Φ _
∂Φ _
∂Φ _ ⎤
∇2 Φ = ∇ i ( ∇Φ ) = ⎢ i x
+ iy
+ iz
ix+
iy+
i ⎥
⎥i⎢
∂x
∂y
∂z ⎦ ⎣ ∂x
∂y
∂z z ⎦
⎣
=
∂2 Φ
∂x2
+
∂2 Φ
∂y2
+
∂2 Φ
∂z2
VI. Coulomb Superposition Integral
1. Point Charge
Er = −
∂Φ
q
q
=
⇒Φ=
+C
2
∂r
4πε0r
4πε0r
Take reference Φ (r → ∞ ) = 0 ⇒ C = 0
Φ=
q
4πε0r
2. Superposition of Charges
6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 4
Page 5 of 6
⎡
⎤
⎢
⎥
q2
dq1
dq2
1 ⎢ q1
+
+ ...
+
+ ...⎥
dΦ T ( P ) =
⎥
4πε0 ⎢ r − r
r − r2
1
r − r'
r − r'
⎢
⎥
1
2
⎣
⎦
Φ T (P ) =
1
4πε0
⎡
⎢ N
qn
⎢
dq
+
⎢
⎢ n =1 r − r n
r − r'
all line,
surface, and
⎢
volume ch arg es
⎣
∑
⎡
N
qn
1 ⎢
⎢
=
+
4πε0 ⎢n =1 r − r
n
⎢⎣
∑
∫
∫
L
λ ⎛⎜ r ' ⎞⎟ dl'
⎝ ⎠
+
'
r −r
∫
S
⎤
⎥
⎥
⎥
⎥
⎥
⎦
σs ⎛⎜ r ' ⎞⎟ da'
⎝ ⎠
+
'
r −r
∫
V
⎤
ρ ⎛⎜ r ' ⎞⎟ dV ' ⎥
⎝ ⎠
⎥
⎥
'
r −r
⎥⎦
Short-hand notation
() ∫
Φ r =
V
ρ ⎛⎜ r ' ⎞⎟ dV '
⎝ ⎠
4πε0 r − r '
6.013, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 4
Page 6 of 6