patterns of magnetic field in space
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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. - PowerPoint PPT PresentationTRANSCRIPT
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)
Will it work for any circular path of radius r ?
IldB 0
Need to compare and11 ldB
22 ldB
||BdlldB
2
||2
1
1
rdl
rdl
1
01
24 r
IB
12
1
2
02
24
Brr
rIB
1
1
21
2
1||2222 dl
rrB
rrdlBldB
1122 ldBldB
A Noncircular Path
𝑑𝑙2∥=𝑟2
𝑟1𝑑𝑙1
Currents Outside the Path
IldB 0
Need to compare and11 ldB
22 ldB
2
||2
1
1
rdl
rdl
12
12 B
rrB
1122 ldBldB
0 ldB
for currents outside the path
101 IldB
202 IldB
03 ldB
pathinsideIldB _0
Ampere’s law
Three Current-Carrying Wires
∮ (𝐵1+𝐵2+𝐵3 ) ∘𝑑 𝑙=𝜇0 ( 𝐼1− 𝐼 2 )
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
Ampere, 1826: Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience
Ampère’s Law
pathinsideIldB _0
1. Choose the closed path2. Imagine surface (‘soap film’) over the path
ldB
3. Walk counterclockwise around the path adding up 4. Count upward currents as positive, inward going as negative
21_ III pathinside uppathinside II _ updownup III
Inside the Path
pathinsideIldB _0
Ampere’s law
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
pathinsideIldB _0
Symmetry: B || path
INrB 02
rNIB 2
40
Is magnetic field constant acrossthe toroid?
Ampere’s Law: A Toroid
0
4110 7 T m
A
What is I ?
A) AB) AC) AD) A
pathinsideIldB _0
What is
B d
rl— ?
A) 0 TmB) 8.7 Tm C) 1.7 TmD) 2.0 Tm E) 2.1 Tm
= .866
, , w=0.5m, h=0.2m,
What is ?