FNT #9: Electric fields and voltages
Damien Martin
November 12, 2005
Question 1: Electric field and electric voltage
The electric field
E
and the voltage
V
at a point
r
=
R
are related by
E
(
r
=
R
) =
vextendsingle
vextendsingle
vextendsingle
vextendsingle
Δ
V
Δ
R
vextendsingle
vextendsingle
vextendsingle
vextendsingle
This tells us that the electric field is related to how quickly the voltage
changes.
Here the “Δ” means we are looking at finite differences in the
voltage and distance, and doing a rise over run. A more elegant and precise
way of doing it is to look at the tangent at the same point:
E
(
r
=
R
) =
vextendsingle
vextendsingle
vextendsingle
vextendsingle
d
V
d
r
vextendsingle
vextendsingle
vextendsingle
vextendsingle
Written in this form, we are reminded that the functional form of
E
can be
obtained from the functional form of
V
via a derivative.
Given that the field of a point change
E
is given by
E
=
k

Q

r
3
find the correct expression for the Voltage from a charge.
Solution
This question is a little bit tricky. To start with, we have that the electric
field is the negative slope of the electric field:
E
=

d
V
d
r
.
This should be given in the problem, but it is in your notes.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
The rest of this problem is realising that to “undo” a derivative we need
to integrate. That is
V
=
integraldisplay
d
V
d
r
d
r.
This is the
fundamental theorem of calculus
. If you have trouble remembering
it, the easiest way is to think of the d
r
on the “denominator” and the d
r
in
the “numerator” cancelling:
integraldisplay
d
V
a26
a26
d
r
a26
a26
d
r
=
integraldisplay
d
V
=
V.
To finish the problem, we need to actually do the integral:
V
=
integraldisplay
d
V
d
r
d
r
=
integraldisplay
(

E
) d
r
=

kQ
integraldisplay
1
r
2
d
r
= +
kQ
r
Question 2: An
α
particle
An
α
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Staff
 Physics, Electric Fields, Electric Potential, Energy, Potential Energy, Particle, Electric charge

Click to edit the document details