Last Lecture
Gauss’s law
Using Gauss’s law for:
spherical symmetry
This lecture
Using Gauss’s law for:
line symmetry
plane symmetry
Conductors in electric fields
Coulomb’s law tutorial:
Consider two positively charged particles, one of charge
q0 (particle 0) fixed at the origin, and another of charge
q1 (particle 1) fixed on the y-axis at (0, d1, 0).
What is the net force F on particle 0 due to particle 1?
Express your answer (a vector) using any or all of k,
q0, q1, d1, x-hat, y-hat, and z-hat.
Example of wrong
answer:
(
Answer: F =
)
Part B
What is the new net force on particle
0, from particle 1 and particle 2?
Answer: F =
Here is an example of a wrong answer - just a careless mistake.
This one presumably meant to put a bracket
around the two terms, but didn’t.
Part D
What is the net force on particle 0
due solely to this charge 3?
Answer: F =
The answer here was close but still
missing a factor of ½.
The message is that accuracy
matters.
A
B
C
D
E
These are two-dimensional cross sections through three
dimensional closed spheres and a cube.
Which of them has the largest flux through surfaces A to E?
Which has the smallest flux? Answers:
ΦB= ΦE
>
ΦA= ΦC= ΦD
The flux through a CLOSED surface depends only on the amount of enclosed charge, not
the size or shape of the surface.
Example 21.2
Field of a hollow spherical sphere
From Gauss’s law, E = 0 inside shell
Example 21.3
Field of a point charge within a shell
Done on board
Read TIP: SYMMETRY MATTERS!
Line symmetry
Example 21.4 Field of a line of charge
A section of an infinitely long
wire with a uniform linear
charge density, .
Find an expression for E at
distance r from axis of wire.

E
20 r
(line of charge)
Applying Gauss’ Law: cylindrical symmetry

E
20 r
(line of charge)
(Compare this result with that
obtained using Coulomb’s law in
Example 20.7, when wire is
infinitely long.)
Applies outside any
cylindrical charge
distribution
Plane symmetry
Example 21.6
Field of an infinite plane
sheet of charge
p 357
Applying Gauss’ Law: planar symmetry
Gaussian surface
(cylinder)
A thin, infinite,
nonconducting sheet with
uniform surface charge
density
Find E at distance r from
sheet.
EA  EA 
qenclosed
0
A
EA  EA 
0

E
2 0
(sheet of charge)
Electric field
due to plane of
charge is

Ex 
2 0
Ex 

2 0
By integration
methods:
Ex  

Ex 
2 0
- which reduces to above
formula for very large R
(see problem 20.69)


x
E  2 k 1 

2
2
x R 


2 0
CHECKPOINT: The figure
shows two large parallel,
nonconducting sheets with
identical (positive) volume
charge density. Rank the
four labelled points
according to the magnitude
of the net electric field
there, greatest first.
A
B
Answer:
C, D equal
B
A
C
D
21.5 Fields of arbitrary charge distributions
CHECKPOINT: There is a certain net flux I through a Gaussian
sphere of radius r enclosing an isolated charged particle.
Suppose the Gaussian surface is changed to
(a) a larger Gaussian sphere,
(b) a Gaussian cube with edge length equal to r, and
(c) a Gaussian cube with edge length 2r.
In each case is the net flux through the new Gaussian
surface
A. greater than
Answers: all equal as
B. less than, or
charge enclosed is
C. equal to I ?
the same
When we must use Coulomb - no symmetry!
Find E at point P due to a finite line
charge
Charge Q is uniformly distributed on a
straight line segment of length L.
Choose axes and draw diagram
Write expression for |dE| due to dq
Find dEy
Integrate to find Ey
(what are the limits?)
Substitute trigonometric formulae
Repeat for Ex
TrueTrue
A or False?
False F
1.
If the net electric flux out of a closed surface is zero, the electric
field must be zero everywhere on the surface.
False
2.
If the net electric flux out of a closed surface is zero, the charge
False
density must be zero everywhere inside the surface.
3.
The electric field is zero everywhere within the material of a
conductor in electrostatic equilibrium.
4.
The tangential component of the electric field is zero at all points
just outside the surface of a conductor in electrostatic
True
equilibrium.
False
The normal component of the electric field is the same at all points
5.
just outside the surface of a conductor in electrostatic
equilibrium.
True