Lecture 2 Notes, 95.657 Electromagnetic Theory I, Fall 2011
Dr. Christopher S. Baird, UMass Lowell
1. Surface Charge Density on an Arbitrary Surface
 Take Gauss's Law in Integral Form
∮
E
⋅
n
da
=
1
0
∫
x
d
3
x
 Where
n
is the vector normal to the bounding surface, and
n
' is the vector normal to the charged surface.
 Choose a small volume that just barely bounds a small
patch on the surface so that the only charge in the volume is
the surface charge:
x
=
n'
−
n'
0
where
n'
is the
component of the position vector in the direction normal to
the charged surface and
n'
0
is the position of the surface in that dimension.
∮
E
⋅
n
d
2
x
=
1
0
∫
n'
−
n'
0
d
3
x
 The surface bounding the volume has three sides: the top, the bottom, and the curved side.
 The curved side can be chosen parallel to the electric fields so that their contribution cancels.
 All that remains is the top and bottom. If the volume is sufficiently small, the patch of charged
surface in the volume can be assumed to be flat so that
n
=
n
' over the top side and
n
= 
n
' over the
bottom side.
∫
top
E
2
⋅
n
' d
2
x
−
∫
bottom
E
1
⋅
n
' d
2
x
=
1
0
∫
n'
−
n'
0
d
3
x
 Also, if the bounding volume is taken small enough, all of the variables can be assumed to be
constant and can be taken out of the integrals:
E
2
⋅
n
'
[
∫
top
d
2
x
]
−
E
1
⋅
n
'
[
∫
bottom
d
2
x
]
=
1
0
[
∫
d
2
x
]
∫
n'
−
n'
0
d
n
'
 All three surfaces are identical and have identical area integrals that cancel out:
E
2
⋅
n
'
−
E
1
⋅
n
'
=
1
0
∫
n'
−
n'
0
d
n
'
 Take the integral of the delta function and drop the primes to simplify notation:
[
(
E
2
−
E
1
)⋅
n
=
1
ϵ
0
σ
]
n
=
n
0
Surface Charge Density on an Arbitrary Surface
 This does not specify the electric field completely and everywhere. It only specifies the normal
component, and only on the surface. It is very useful however and will be used again and again. It
n
'
E
2
E
1
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is essential that you master this derivation. It is the first general boundary condition of Maxwell's
equations. We also need a similar equation that will link the tangential components of the electric
field along a boundary.
 This equation can be used for the following:
1. To find the surface charge density if we know the fields
2. To connect two regions at their boundary if we know the fields in both regions up to a
constant
3. To find the fields everywhere if we know the surface charge density and the geometry is
especially simple.
2. The Scalar Potential
 Gauss's law involves the electric field, which is a vector property. Because solving vector
equations is much more difficult and involved than solving scalar equations, the mathematics can
be simplified by transforming Gauss's law into a scalar form.
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 Fall '11
 Staff
 Charge, Electric Potential, Electrostatics, Mass, Electric charge

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