Electric Fields of Charge Distribution
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Electrons on an insulator (zoomed in).
Negative…
Zoomed out – difficult to “see” individual
electrons ≈ smooth distribution of charge.
 Like charge is “smeared” on the insulator
𝐸 = lim 𝑘𝑒
Δ𝑞𝑖 →0
𝐸 = 𝑘𝑒
𝑖
Δ𝑞𝑖
2 𝑟𝑖
𝑟𝑖
𝑑𝑞
𝑟,
2
𝑟
where integral is over the
entire distribution. Can
also write:
𝑑𝑞
𝐸 = 𝑘𝑒
𝑟.
3
𝑟
Different Distributions
Small Charge
charge density
Small
length
linear charge density, 𝜆: 𝑑𝑞 = 𝜆 𝑑𝑙
surface charge density, 𝑑𝑞 = 𝜎 𝑑𝐴
Small
area
volume charge density, 𝑑𝑞 = 𝜌 𝑑𝑉
Small
volume
Solution Process for 𝑬, p.672
1) Visualize the electric field from the
distribution and/or individual charges. Note
any symmetries.
2) Categorize: individual charges or distribution?
3) Analyze:
a) Individual charges – sum.
b) Distribution – integral (see table: Appendix B,
Page A-18.)
4) Check: does the result match with your
visualization at the beginning? Is it symmetric
in ways that it should be?
All charged rods have the same length and the same linear
charge density (+ or -). Light rods are positively charged, and
dark rods are negatively charged. For which arrangement
below would the magnitude of the electric field at the origin
be largest?
1.
2.
3.
4.
5.
6.
7.
8.
Impossible to
determine
Electric Field Lines
Helpful for visualizing electric fields.
In the same direction as 𝐸 at each point in
space.
Direction of force on a positive test charge
placed there  direction of acceleration for
positive charge.
Where does the negatively charged triangle go?
Rules for Sketching
1. Lines begin on positive charges
terminate on negative charges
and
– If more of one type of charge, some lines will
begin or end infinitely far away.
2. The number of lines leaving a positive charge
or approaching a negative charge is
proportional to the magnitude of the charge.
3. No two field lines can cross.
May be helpful as you
practice this:
http://phet.colorado.ed
u/sims/charges-andfields/charges-andfields_en.html
Two uniformly charged rods are positioned
horizontally as shown. The top rod is positively
charged and the bottom rod is negatively charged.
The total electric field at the origin:
1.
2.
3.
4.
5.
6.
7.
is zero.
has both a non-zero x component
and a non-zero y component.
points totally in the +x direction.
points totally in the –x direction.
points totally in the +y direction
points totally in the –y direction.
points in a direction impossible to
determine without doing a lot of
math.
Rank the magnitudes of the electric field at points A, B,
and C shown in the figure(greatest magnitude first).
1.
2.
3.
4.
5.
6.
A, B, C
C, A, B
B, C, A
C, B, A
A, C, B
B, A, C
Motion of a Charged Particle
Newton’s 2nd Law: ∑𝐹 = 𝑚𝑎
𝐸
𝑎=𝑞
𝑚
If 𝐸 is “uniform” (read: constant throughout
space)  kinematics.
Example: A uniformly charged rod of length L
and total charge Q lies along the x axis as shown
in the figure. (a) Find the components of the
electric field at the point P on the y axis a
distance d from the origin. (b) What are the
approximate values of the field components
when d >> L? Explain why you would expect
y
these results.
P
d
L
x
Quantitative Question:
You are helping to design a new electron
microscope to investigate the structure of the HIV
virus. A new device to position the electron beam
consists of a charged circle of conductor. This circle
is divided into two half circles separated by a thin
insulator so that half of the circle can be charged
positively and half can be charged negatively. The
electron beam will go through the center of the
circle. To complete the design your job is to
calculate the electric field in the center of the circle
as a function of the amount of positive charge on
the half circle, the amount of negative charge on
the half circle, and the radius of the circle.
Lab: None
Start on homework:
P37, P42, P43, P46 from Chapter 23