Download Ch. 22.1-22.7 revisited

Chapter 22
Patterns of Fields in
Space
• Electric flux
• Gauss’s law
• Ampere’s law
• Maxwell equations
Patterns of Fields in Space
What is in the box?
no charges?
vertical charged plate?
Patterns of Fields in Space
Box versus open surface
Seem to be able to tell
if there are charges inside
…no clue…
Gauss’s law:
If we know the field distribution on closed surface
we can tell what is inside.
Electric Flux: Surface Area
3. Surface area
flux through small area:

flux ~ E  nˆA
Definition of electric flux on a surface:

 E  nˆA
surface
The Electric Field of a Large Plate
Symmetry:
Field must be perpendicular to surface
Eleft=Eright

 E  nˆA 
q
inside
0
surface
2EAbox
Q / AAbox


0
Q / A

E
2 0
The Electric Field of a Uniform Spherical Shell of
Charge
Symmetry:
1. Field should be radial
2. The same at every location
on spherical surface

 E  nˆA 
q
inside
0
surface
A. Outer sphere:
Q
2
E 4r 
0
B. Inner sphere:
0
2
E 4r 
0
1
Q
E
40 r 2
E 0
Gauss’s Law: Properties of Metal
Can we have excess charge inside a metal that is in
static equilibrium?
Proof by contradiction:

 E  nˆA 
q
surface
=0
q
inside
0
0
inside
0
Gauss’s Law: Hole in a Metal

 E  nˆA 
surface
=0
q
inside
0
q
inside
0
0
What is electric field inside?
VACB =
0
VADB
 
   E  dl  0
1. No charges on the surface of an
empty hole
2. E is zero inside a hole
Gauss’s Law: Charges Inside a Hole

 E  nˆA 
q
surface
=0
inside
0
q
inside
0
+5nC
q
 qinside 0
q
 5 nC
surface
surface
0
Gauss’s Law: Screening

 E  nˆA 
surface
q
inside
0
Is the field zero inside the box because the metal blocks the
field?
Gauss’s Law: Circuits
Can we have excess charge inside in steady state?

 E  nˆA 
surface

 E  nˆA  
left _ surface

 E  nˆA
right_ surface
q
inside
0
q
inside
0
0
Gauss’s Law: Junction Between Two Different
Metal Wires
n2<n1
u2<u1
i1=i2
n1Au1E1 = n2Au2E2
n1u1
E2 
E1  E1
n2u2

 E  nˆA 
surface
q
inside
q
inside
0
  0 (E1 A  E2 A)  0
There is negative charge along the interface!
Magnet Cut in Half & Pulled Apart
No magnetic monopole!
Try to cut a magnet down to a single pole, just get smaller magnets
No magnetic Charge!
Gauss’s Law for Magnetism
Dipoles:
Electric field: ‘+’ and ‘–’ charges can be separated
Magnetic field: no monopoles
Suppose magnetic dipole consists of two magnetic monopoles,
each producing a magnetic field similar to the electric field.
One cannot separate them  total magnetic ‘charge’ is zero.

 E  nˆA 
surface
q
inside
0

 B  nˆA  0
surface
or

 B  nˆA  0
Gauss’s law for magnetism
Patterns of Magnetic Field in Space
Is there current passing
through these regions?
There must be a relationship between the
measurements of the magnetic field along a closed
path and current flowing through the enclosed area.
Ampere’s law
Quantifying the Magnetic Field Pattern
0 2 I
Bwire 
4 r
 
Curly character – introduce:  B  dl
  0 2 I
0 2 I
 B  dl  4 r  dl  4 r 2r
 
 B  dl  0 I
Similar to Gauss’s law (Q/0)
Ampère’s Law
 
 B  dl  0  Iinside_ path
All the currents in the universe contribute to B
but only ones inside the path result in nonzero path integral
Ampere’s law is almost equivalent to the Biot-Savart law:
but Ampere’s law is relativistically correct
Ampere’s Law: A Long Thick Wire
 
 B  dl  0  Iinside_ path
Can B have an out of plane component?
Is it always parallel to the path?
 
 B  dl  B2r
B2r  0 I
for thick wire:
0 2 I
B
4 r
(the same as for thin wire)
Would be hard to derive using Biot-Savart law
Ampere’s Law: An Infinite Sheet
 
 B  dl  0  Iinside_ path
Number of wires inside: 𝑑(N/L)
 
What is B  dl on sides?
𝐵
𝐵
𝑑
Each wire has 𝐼
𝐵𝑑 + 𝐵𝑑 = 𝜇0 𝑑(𝑁 𝐿)𝐼
𝜇0 𝑁𝐼
𝐵=
2𝐿
Uniform: Does not depend on distance from sheet.
Opposite directions above and below sheet.
Ampere’s Law: A Toroid
 
 B  dl  0  Iinside_ path
Symmetry: B || path
B2r  0 IN
0 2 NI
B
4 r
Is magnetic field constant across
the toroid?
Maxwell’s Equations
Three equations:
Gauss’s law for electricity

 E  nˆdA 
q
inside
0

Gauss’s law for magnetism
 B  nˆA  0
 
Ampere’s law for magnetism  B  dl  0  I inside_ path
Is anything missing?
 
‘Ampere’s law for electricity’  E  dl  0
(incomplete)
Maxwell’s Equations (incomplete)

 E  nˆdA 
q
inside
0
Gauss’s law for electricity

Gauss’s law for magnetism
 B  nˆA  0
 
Incomplete version of Faraday’s law
 E  dl  0
 
Ampere’s law
B

d
l


I

0
inside_ path

(Incomplete Ampere-Maxwell law)
First two: integrals over a surface
Second two: integrals along a path
Incomplete: no time dependence
Motional EMF Revisited
Heat Ring and it expands
𝑣
𝐸
Ampere’s law
B
𝐸
What is the direction of
the electric field on the
ring?
Curly electric fields!
What is changing inside
the ring?
𝑑𝜙𝐵
𝑒𝑚𝑓 = −
𝑑𝑡