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Dr. Gregory J. Mazzaro
Spring 2017
ELEC 318 – Electromagnetic Fields
Lecture 4(d)
Electrostatic
Boundary Conditions
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Boundaries between Material Media
boundary conditions:
-- refer to the behavior of fields (E, H)
and flux densities (D, B) at the surfaces
where material media meet
-- electrostatic media are characterized by
e = permittivity (dielectric constant)
s = conductivity
-- in general, across a boundary, E1  E2
For a boundary between two media given in spherical coordinates by R = 3 m ,
determine the components of E which are normal and tangential to the boundary
at P(3, p/2, p/4) if
ˆ  2sin   θˆ  6cos  4  ˆ V m
E  4R R
V
ˆ V
Et  2 θˆ  6 ˆ
, En  12 R
m
m
2
Tangential Electric Field Intensity
E  E1t  E1n
w
h 2
To describe the behavior of
electric fields tangential to
the boundary, we use the fact
that electrostatic fields are
irrotational :

abcd
w
h 2
E  dl  0
E2t w  E2 n  h 2   E2 n  h 2   E1t w  E2 n  h 2   E2 n  h 2   0
E2t w  E1t w  0

E1t  E2t
 Across a boundary between material media, tangential electric field intensity is continuous.
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Normal Electric Flux Density
D  eE
To describe the behavior of electric fields
normal to the boundary, we use Gauss’ Law:

S
n̂
D1
D1n
D  dS  Qenc
D1t
D2t
s = charge per unit area, at the boundary
D2n
S
h 2
s
h 2
D2
Qenc  s S
nˆ
 D1n  nˆ S    D2n  nˆ S 
 s
D1n  D2 n   s
 For a charge-free boundary (s = 0), normal electric flux density is continuous.
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Example: Charge-Free Boundary
With reference to this figure (at right), determine
E1 if E2 is given by
E2  5xˆ  9yˆ  3zˆ V m
and the two materials are characterized by
e1  2e 0 , e 2  8e 0
Assume that s = 0 at the boundary.
E1t  E2t , D1n  D2n  s
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Example: Charge at the Boundary
With reference to this figure (at right), determine
E1 if E2 is given by
E2  5xˆ  9yˆ  3zˆ V m
and the two materials are characterized by
e1  2e 0 , e 2  8e 0
Assume that s = 35.4 pC/m2 at the boundary.
E1t  E2t , D1n  D2n  s
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Example: Dielectric/Conductor
With reference to this figure (at right), determine
E1 and E2 if s = 35.4 pC/m2 at the boundary
and material 1 has a dielectric constant of er1 = 2 ,
and material 2 is a perfect conductor .
conductor
E1t  E2t , D1n  D2n  s
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Dr. Gregory J. Mazzaro
Spring 2017
ELEC 318 – Electromagnetic Fields
Lecture 4(x)
Electrostatic Fields:
Additional Examples
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Example: Linear Superposition
Three infinitely-long lines of charge, each with density 445 pC/m, are parallel to the z axis.
One is on the z-axis (x = 0, y = 0). The second is at x = 0, y = –3 m.
The third is at x = 0, y = 3 m.
Determine the electric field intensity at P(x = 4 m, y = 3 m, z = 6 m),
in free space.
Prior result: For a single line charge… E 
l
rˆ
2pe 0 r
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Example: Surface Charge, Coulomb’s
z
Calculate the electric field intensity at any point P above
an infinite sheet of constant charge density S in the x-y plane
by calculating the field along the z-axis for a disc of radius R0
and taking the limit of this result as R0  
y
x
rˆ  cos  xˆ  sin  yˆ
ˆ   sin  xˆ  cos  yˆ
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Example: Surface Charge, Gauss’ Law
z
Calculate the electric field intensity at any point P above
an infinite sheet of constant charge density S in the x-y plane
using Gauss’ Law.
Q

S
D  dS
y
x
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