chapter 22 patterns of fields in space electric flux gauss’s law ampere’s law maxwell equations
TRANSCRIPT
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
3. Surface area
flux through small area:
AnEflux ˆ~
Definition of electric flux on a surface:
surface
AnE ˆ
Electric Flux: Surface Area
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 2
04
1
r
QE
B. Inner sphere:
0
2 04
rE 0E
The Electric Field of a Uniform Spherical Shell of Charge
Can we have excess charge inside a metal that is in static equilibrium?
Proof by contradiction:
0
ˆ
inside
surface
qAnE
=0
00
insideq
Gauss’s Law: Properties of Metal
0
ˆ
inside
surface
qAnE
=0
00
insideq
What is electric field inside?
0 ACBV
0ldEVADB
=
1. No charges on the surface of an empty hole
2. E is zero inside a hole
Gauss’s Law: Hole in a Metal
+5nC
0
ˆ
inside
surface
qAnE
=0
00
insideq
0 insidesurface qq
nC 5 surfaceq
Gauss’s Law: Charges Inside a Hole
0
ˆ
inside
surface
qAnE
Gauss’s Law: Screening
Is the field zero inside the box because the metal blocks the field?
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 magnetism
0ˆ 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
r
IBwire
2
40
Curly character – introduce: ldB
dlr
IldB
2
40
rr
I
22
40
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
r
IB
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 inside: (N/L)
What is on sides? ldB
Uniform: Does not depend on distance from sheet. Opposite directions above and below sheet.
Ampere’s Law: An Infinite Sheet
Each wire has 𝐼𝑑
𝐵
𝐵𝑑+𝐵𝑑=𝜇0𝑑 ¿
𝐵
𝐵=𝜇0 𝑁 𝐼
2𝐿
pathinsideIldB _0
Symmetry: B || path
INrB 02
r
NIB
2
40
Is magnetic field constant acrossthe toroid?
Ampere’s Law: A Toroid
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
Motional EMF Revisited
Ampere’s lawB
Heat Ring and it expands
What is the direction of the electric field on the ring?
𝑣
Curly electric fields!
What is changing inside the ring?
𝑒𝑚𝑓=−𝑑𝜙𝐵
𝑑𝑡
𝐸
𝐸