This is an animated derivation for the electric field at a distance H above one end of a finite line of uniformly distributed electric charge.

1
This is an animated derivation for the electric
field at a distance H above one end of a finite
line of uniformly distributed electric
charge. The idea of breaking up the line of
charge into many (ultimately an infinite number)
small elements is presented the mathematical
representation of the field due to one
representative infinitesimal element is
emphasized so that integration can be used to add
up all of the different field components. The
narrator will have to explain the transition from
the demonstrated finite number of finite elements
to an infinite number of infinitesimal
elements. I have found this most useful when the
students are given the chance to do the work
before I show the steps. This requires a bit of
coaching as well as the usual lecture-narration.
Best wishes, Leo Takahashi, The Pennsylvania
State University, Beaver Campus
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y
(0,H)
H
Total Charge Q ?L
?
(DL,0)
(D,0)
(0,0)
x
L
D
3
Total Charge Q ?L
y
dE
(0,H)
dq ?dx
H
r
dq
?
(DL,0)
(D,0)
(0,0)
x
L
D
x
4
y
dE
q
(0,H)
r
H
q
dq
x
(0,0)
L
D
x
dE -dEx i dEy j
dE - dECosq i dESinq j
E ? dE - ? dECos?i ? dESin?j
dE kdq/r2
dq ?dx
5
y
dE -dEx i dEy j dE - dECosq i dESinq j
dE
q
(0,H)
r
H
q
dq
x
(0,0)
L
D
x
(DL,0)
r2 x2 H2
Cos? x/r Sin? H/r
6
dE
q
(0,H)
r
H
q
dq
x
(0,0)
L
D
x
(DL,0)
dECos? (k?dx/r2)(x/r) k?xdx/r3
dECos? k?xdx/(x2 H2)3/2
dESin? (k?dx/r2)(H/r) k?Hdx/r3
dESin? k?Hdx/(x2 H2)3/2
7
E -Ex i Ey j
E ? dE - ? dExi ? dEyj
E ? dE - ? dECos?i ? dESin?j
dECos? k?xdx/(x2 H2)3/2
dESin? k?Hdx/(x2 H2)3/2
8
E -Ex i Ey j
y
(0,H)
H
?
(DL,0)
(D,0)
x
(0,0)
L
D