Lecture 5: Electric fields and electric flux
[Previous lecture: Electric field of a point charge]
𝑞
𝐸 = 𝑘 ! 𝒓&
𝑟
[Previous lecture: Electric field of multiple point charges]
𝐸 = 𝐸" + 𝐸! + 𝐸#
𝑘𝑞"
𝑘𝑞!
𝑘𝑞#
= ! 𝒓& 𝟏 + ! 𝒓& 𝟐 + ! 𝒓& 𝟑
𝑟"
𝑟!
𝑟#
[Previous lecture: You can divide any object into a bunch of “point charges”]
Add up the electric field
from each little piece
𝑘𝑞!
𝐸 = # " 𝒓' 𝒊
𝑟!
!
In the limit where each
piece is infinitesimally
small:
𝑑𝑞
𝐸 = ∫ 𝑘 " 𝒓'
𝑟
[Previous lecture: a line of charge]
What is 𝐸 at point A?
𝐿/2
𝐿/2
s
A
The line has a charge per unit
length −𝜆 (units of C/m)
(Note: I changed the
distance that was
called “d” in lecture to
be called “s”, to avoid
confusion in notation)
Before you do any integrals:
Decide which way 𝐸 needs to point.
[Previous lecture: Electric field from a ring of charge]
What is 𝐸 at point P?
𝜆 = charge per unit length
(units of C/m)
Total charge is 𝑄 = 𝜆×2𝜋𝑅
(and sits in the x-y plane)
Before you do any integrals:
Decide which way 𝐸 needs to point.
Example: a really big disk of charge
What is 𝐸 at point P?
𝜎 = charge per unit area
(units of C/m2)
[this solution is provided for your cultural edification]
What is 𝐸 at point P?
𝜎 = charge per unit area
(units of C/m2)
An introduction to Gauss’s Law,
starting with point charges:
+ charges are “sources” of 𝐸
+ charges are “sinks” of 𝐸
?
You can tell whether a
region of space
contains + or – charge
by checking whether
electric field lines
come in or go out
Let’s play a game:
What is the
total charge
inside the
box?
A) +
B) C) Zero
?
What is the
total charge
inside the
box?
A) +
B) C) Zero
?
What is the
total charge
inside the
box?
A) +
B) C) Zero
-2
?
+1
What is the
total charge
inside the
box?
?
A) +
B) C) Zero
What is the
total charge
inside the
box?
A) +
B) C) Zero
?
Gauss’s law
𝑄./01
= Φ3
𝜖2
You can tell how much charge is inside any region by looking at the
electric flux into / out of that region
What is electric flux?
Φ3 = % 𝐸4 ⋅ 𝐴⃗4
i
What is the flux through this piece of surface?
Integral definition of electric flux
Φ3 = ∫ 𝐸 ⋅ 𝑑𝐴
Electric field has a
constant magnitude 𝐸
and points in the
vertical direction
Shape has a
square cross
section
Which sides of this shape have nonzero
electric flux?
What is the value of the total electric
flux Φ0 across the surface?
The electric field next to a disk of charge,
revisited
How can we quickly find out the magnitude
of the electric field 𝐸 above the disk?
𝑅→∞
Use Gauss’s law to figure out the value of the electric field
Charge 𝜎 per
unit area