Potential on axis of square
The Problem:
Given a r surface with charge density
σ
and edge length
2
L
, find the electric potential at a point a distance
z
perpendicularly away from the center of the square.
A Solution:
Chop & Coordinates
Slice plate into many thin rods of length 2
L
.
Choose
x
and
y
axes in the plane of the square and centered on it, with the
y
axis parallel to the
thin rods.
That way, the thin rods have width
dx
.
The
z
axis passes through the point of interest.
Contribution
An arbitrary rod contributes a potential of
22
ln
Ld L
Vk
λ
⎛⎞
++
=
⎜⎟
+−
⎝⎠
, where
d
is the distance perpendicularly from the midpoint of the rod
to the point of interest.
In this case,
d
is slanted, connecting a point on the
x
axis with the point
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 Spring '08
 IYER
 Physics, Charge, Electric Potential, Euclidean geometry, ln dx L2

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