Download Lecture 24 - Purdue University

Chapter 22
Patterns of Fields in
Space
• Electric flux
• Gauss’s law
• Ampere’s law
• Maxwell equations
Gauss’s Law

 E  nˆA 
surface
q
inside
0

 E  nˆdA 
q
inside
0
Can derive one from another
Gauss’s law is more universal:
works at relativistic speeds
1
Q
E
40 r 2
Gauss’s law:
If we know the field distribution on closed surface
we can tell what is inside.
1. Knowing E can conclude what is inside
2. Knowing charges inside can conclude what is E
Clicker
Q2
A solid conducting sphere of radius rA has charge
Q1 uniformly distributed over its surface.
Q1
rA
rB
rC
Concentric with the solid sphere is a conducting
shell of with charge Q2 on its outer surface.
What is the charge Q on the inner surface of the
outer shell?
A.
B.
C.
D.
E.
0
+Q2
-Q2
+Q1
-Q1
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
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)
Will it work for any circular path of radius r ?
A Noncircular Path (home study)
 
 B  dl  0 I
 
 
Need to compare B1  dl1 and B2  dl2
 
0 2 I
B1 
B  dl  Bdl||
4 r1
dl1 dl2||

r1
r2
0 2 I r1
B2 
 B1
4 r2 r2
 
 r1  r2 
B2  dl2  B2dl2||   B1  dl1 
 r2  r1 
   
B2  dl2  B1  dl1
Currents Outside the Path (home study)
 
 B  dl  0 I
 
 
Need to compare B1  dl1 and B2  dl2
dl1 dl2||

r1
r2
r1
B2  B1
r2
 
 
B2  dl2   B1dl1
 
 B  dl  0
for currents outside the path
Three Current-Carrying Wires
 
 B1  dl  0 I1
 
 B2  dl  0 I 2
 
 B3  dl  0
Ampere’s law
 
 B  dl  0  Iinside_ path
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, 1826: Memoir on the Mathematical Theory of Electrodynamic
Phenomena, Uniquely Deduced from Experience
Maxwell: We can scarcely believe that Ampère really discovered the law of action by means of the experiments
which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some
process which he has not shown us, and that when he had afterwards built up a perfect demonstration he
removed all traces of the scaffolding by which he had raised it.
Inside the Path
Ampere’s law
 
 B  dl  0  Iinside_ path
1. Choose the closed path
2. Imagine surface (‘soap film’) over the path
 
3. Walk counterclockwise around the path adding up  B  dl
4. Count upward currents as positive, inward going as negative
(right hand rule: 4 fingers along the path then thumb is positive current)
I
inside_ path
 I1  I 2
I
inside_ path
 I up
 I up  I down  I up
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
Clicker
What is the magnetic field inside a wire
(wire radius is R) at distance r1 from its
center?
A. B=20r1I/(R2)
B. B=r1I/(20R2)
C. B=0r1I/2R2
D. B=0R2I/2r1R2
E. 0
R
r1
Ampere’s Law: find B inside a solenoid
 
 B  dl  0  Iinside_ path
Number of wires: (N/L)d
 
What is B  dl on sides?
B outside is very small
 
 B  dl  Bd
Bd  0 I N / L d
B
0 IN
L
Uniform: same B no matter where is the path
(solenoid)
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
Maxwell’s Equations
Three equations:
Gauss’s law for electricity

 E  nˆdA 
q
inside
0
Gauss’s law for magnetism
 
Ampere’s law for magnetism  B  dl  0  I inside_ path
Is anything missing?
 
‘Ampere’s law for electricity’  E  dl  0
First two: integrals over a surface
Second two: integrals along a path
Incomplete: no time dependence
(incomplete)