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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
=
λdx
and for circular line charges:
dq
=
λR 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:
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This note was uploaded on 12/05/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.
 Spring '08
 Any
 Physics, Charge

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