Download Coulomb`s Law for Charge Density

10/19/2004
Coulombs Law for Charge Density.doc
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Coulomb’s Law for
Charge Density
Consider the case where there are multiple point charges
present. What is the resulting electrostatic field ?
r1′
r2′
+
r
+ Q1
Q2
E (r )
The electric field produced by the charges is simply the vector
sum of the electric field produced by each (i.e., superposition!):
E (r ) =
Q1 r-r1′
Q2 r-r2′
+
4πε 0 r-r ′ 3 4πε 0 r-r ′ 3
1
2
Or, more generally, for N point charges:
Qn r-rn′
E (r ) = ∑
3
n =1 4πε 0 r-r ′
N
n
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/19/2004
Coulombs Law for Charge Density.doc
2/3
Consider now a volume V that is filled with a “cloud” of charge,
descirbed by volume charge density ρv ( r ) .
A very small differential volume
dv, located at point r′ , will thus
contain charge dQ = ρv ( r′ ) dv ′ .
This differential charge
produces an electric field at
point r equal to :
dE ( r ) =
r′
dv ′
r
ρv
E (r )
ρv ( r′ ) dv ′ r-r′
3
4πε 0
r-r′
The total electric field at r (i.e., E ( r ) ) is the summation (i.e.,
integration) of all the electric field vectors produced by all the
little differential charges dQ that make up the charge cloud:
E (r ) =
∫∫∫
V
ρv ( r′ ) r-r′
dv ′
4πε 0 r-r′ 3
Note: The variables of integration are the primed coordinates,
representing the locations of the charges (i.e., sources).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/19/2004
Coulombs Law for Charge Density.doc
3/3
Similarly, we can show that for surface charge:
E (r ) =
∫∫S
ρs ( r′ ) r-r′
ds ′
4πε 0 r-r′ 3
∫
ρ A ( r′ ) r-r′
d A′
4πε 0 r-r′ 3
And for line charge:
E (r ) =
C
Jim Stiles
The Univ. of Kansas
Dept. of EECS