Download - Chapter 22
Chapter 22
Patterns of Fields in Space
• Electric flux• Gauss’s law• Ampere’s law• Maxwell equations
What is in the box?
no charges? vertical charged plate?
Patterns of Fields in Space
Box versus open surface
Seem to be able to tellif there are charges inside
…no clue…
Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.
Patterns of Fields in Space
0
ˆ
inside
surface
qAnE
0
ˆ
insideqdAnE
Symmetry makes it simple!
Gauss’s Law
Symmetry: Field must be perpendicular to surfaceEleft=Eright
0
ˆ
inside
surface
qAnE
2EAbox Q / A Abox
0
E Q / A 20
The Electric Field of a Large Plate
Symmetry: 1. Field should be radial2. The same at every location
on spherical surface
0
ˆ
inside
surface
qAnE
A. Outer sphere:
0
24
QrE 204
1rQE
B. Inner sphere:
0
2 04
rE 0E
The Electric Field of a Uniform Spherical Shell of Charge
0
ˆ
inside
surface
qAnE
Is Gauss’s law still valid?
Can we find E using Gauss’s law?
The Electric Field of a Uniform Cube
Gauss’s Law for Electric Dipole
Can we have excess charge inside in steady state?
0
ˆ
inside
surface
qAnE
surfacerightsurfaceleft
AnEAnE__
ˆˆ
00
insideq
Gauss’s Law: Circuits
Gauss’s Law: Junction Between Two Different Metal Wires
i1=i2
n1Au1E1 = n2Au2E2
E2 n1u1
n2u2
E1 E1
0
ˆ
inside
surface
qAnE
There is negative charge along the interface!
qinside 0 (E1A E2A) 0
n2<n1u2<u1
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!
Dipoles:Electric field: ‘+’ and ‘–’ charges can be separatedMagnetic 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.
0
ˆ
inside
surface
qAnE
Gauss’s law for magnetism0ˆ
surface
AnB
0ˆ AnBor
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
rIBwire
24
0
Curly character – introduce: ldB
dlrIldB 2
40
rrI
224
0
IldB 0
Similar to Gauss’s law (Q/0)
All the currents in the universe contribute to Bbut 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
Ampère’s Law
pathinsideIldB _0
pathinsideIldB _0
Can B have an out of plane component?
Is it always parallel to the path?
rBldB 2
IrB 02
rIB 2
40
for thick wire: (the same as for thin wire)
Would be hard to derive using Biot-Savart law
Ampere’s Law: A Long Thick Wire
pathinsideIldB _0
Number of wires: (N/L)d
What is on sides? ldB
B outside is very small
BdldB
Bd 0I N / L dLINB 0
(solenoid)
Uniform: same B no matter where is the path
Ampere’s Law: A Solenoid
Three equations:
Gauss’s law for electricity
Gauss’s law for magnetism
Ampere’s law for magnetism pathinsideIldB _0
0
ˆ
insideqdAnE
Is anything missing?
‘Ampere’s law for electricity’ ldE
0 ldE
(incomplete)
Maxwell’s Equations
0ˆ AnB
0
ˆ
insideqdAnE
0 ldE
pathinsideIldB _0
Gauss’s law for electricity
Gauss’s law for magnetism
Incomplete version of Faraday’s law
Ampere’s law(Incomplete Ampere-Maxwell law)
First two: integrals over a surfaceSecond two: integrals along a path
Incomplete: no time dependence
Maxwell’s Equations (incomplete)
0ˆ AnB