PHYSICS –PHY101 VU
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cosdE
θ
dE
G
θ
θ
R
x
y
z
ds
λ
P
Summary of Lecture 23 – ELECTROSTATICS II
1. In the last lecture we learned how to calculate the electric field if there are any number
of point charges. But how to calculate this when charges are continuously distributed over
some reg
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ion of space? For this, we need to break up the region into little pieces so that
each piece is small enough to be like a point charge. So, , or
is the total el
i
EE E E
EE
=Δ +Δ +Δ +⋅⋅⋅⋅
=Δ
∑
GG G G
GG
ectric field. Remember that is a vector that can be resolved
ˆ
ˆˆ
into components, . In the limit where the pieces are small enough,
we can write it as an integral, (or
xy z
E
EEiEjEk
EdE
=+ +
=∫
G
G
GG
, , )
2. Charge Density: when the charges are continuously distributed over a region - a line, the
surface of a material, or inside a sphere - we must specify the
xxyyzz
EdEEdEEdE
charge de
===
∫∫∫
. Depending
upon how many dimensions the region has, we define:
(a) For linear charge distribution:
(b) For surface charge distribution:
nsity
dq ds
dq dA
λ
σ
=
=
(c) For volume charge distribution:
The dimensions of , , are determined from the above definitions.
3. As an example of how we work out the electric field coming from a continuous
dq dV
ρ
λσ ρ
=
charge
distribution, let us work out the electric field from a uniform ring of charge at the point P.
()
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00
The small amount of charge gives rise to an electric field whose magnitude is
1
44
The component in the z direction is cos
z
ds
ds ds
dE rzR
dE dE
λ
λλ
πε πε
θ
==
+
=
()
1/2
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with cos .
zz
rzR
θ
== +
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