3.3
The Electric Field for Continuous Charge Distributions
For a continuous charge distribution, we can write down the differential bit of electric field due to
a differential bit of charge as:
dE
=
1
4
π
0
dq
r
2
(6)
Using this knowledge, we can find the total electric field by summing up (integrating) all of the
contributions from the differential elements of charge.
The challenge in doing so usually comes
down to expressing the charge in a way such that it can be integrated over.
For line charges we can use the linear charge density to express
dq
in terms of an integration
variable. For straight line charges:
dq
=
λ dx
and for circular line charges:
dq
=
λ R dθ
Using the second form, we can derive the electric field due to a ring of charge as measured on the
axis of the ring:
E
=
1
4
π
0
qz
(
z
2
+
R
2
)
3
/
2
(7)
Similarly, for a charged surface such as a disk or a plane, we can use the surface charge density
σ
to express
dq
in terms of an integration variable. In the case of a charged disk:
dq
=
σ dA
=
σ
(2
πr
)
dr
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 Spring '08
 Any
 Physics, Charge, Electric charge, Fundamental physics concepts

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