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Physics 2102 Magnetic fields produced by
currents
Physics 2102Gabriela González
The Biot-Savart Law
• Quantitative rule for computing the magnetic field from any electric current
• Choose a differential element of wire of length dL and carrying a current i
• The field dB from this element at a point located by the vector r is given by the Biot-Savart Law:
Ld
r
30
4 r
rLidBd
i
m0 =4px10-7 T.m/A(permeability constant)
Jean-Baptiste Biot (1774-1862)
?Felix Savart (1791-1841)
Compare with:3
0
1
4
dq rdE
r
Biot-Savart Law• An infinitely long straight wire
carries a current i. • Determine the magnetic field
generated at a point located at a perpendicular distance R from the wire.
• Choose an element ds as shown• Biot-Savart Law:
dB ~ ds x r points INTO the page
• Integrate over all such elements3
0
4 r
rsidBd
30 )sin(
4 r
ridsdB
3
0 )sin(
4 r
rdsiB
2/322
0
4 Rs
Rdsi
02/322
0
2 Rs
Rdsi
0
2/1222
0
2 RsR
siR
R
i
2
0
rR /sin 2/122 )( Rsr
3
0 )sin(
4 r
rdsiB
Field of a straight wire (continued)
A Practical Matter
A power line carries a current of 500 A. What is the magnetic field in a house located 100m away from the power line?
R
iB
2
0
)100(2
)500)(/.104( 7
m
AAmTx
= 1 mT!!
Recall that the earth’s magnetic field is ~10-4T = 100 mT
Biot-Savart Law• A circular loop of wire of
radius R carries a current i. • What is the magnetic field at
the center of the loop?
ds
R
30
4 r
rdsiBd
20
30
44 R
iRd
R
idsRdB
R
i
R
i
R
idB
24
2
4000
Direction of B?? Not another right hand rule?!Curl fingers around direction of CURRENT.Thumb points along FIELD! Into page in this case.
i
aI
B
2
101
Magnetic field due to wire 1 where the wire 2 is,
1221 BILF
a
I2I1L
aIIL
2210
Force on wire 2 due to this field,F
Forces between wires
Given an arbitrary closed surface, the electric flux through it isproportional to the charge enclosed by the surface.
qFlux=0!
q
Surface 0
qAdE
Ampere’s law: Remember Gauss’ law?
Gauss’ law for magnetism:simple but useless
No isolated magnetic poles! The magnetic flux through any closed “Gaussian surface” will be ZERO. This is one of the four “Maxwell’s equations”.
0AdB
0q
AdE
Always! No isolated magnetic charges
IsdB 0loop
The circulation of B (the integral of B scalards) along an imaginary closed loop is proportionalto the net amount of currenttraversing the loop.
i1
i2i3
ds
i4
)( 3210loop
iiisdB
Thumb rule for sign; ignore i4
As was the case for Gauss’ law, if you have a lot of symmetry,knowing the circulation of B allows you to know B.
Ampere’s law: a useful version of Gauss’ law.
Ampere’s Law: Example• Infinitely long straight wire
with current i.• Symmetry: magnetic field
consists of circular loops centered around wire.
• So: choose a circular loop C -- B is tangential to the loop everywhere!
• Angle between B and ds = 0. (Go around loop in same
direction as B field lines!)
C
isdB 0
C
iRBBds 0)2(
R
iB
2
0
R
Ampere’s Law: Example
• Infinitely long cylindrical wire of finite radius R carries a total current i with uniform current density
• Compute the magnetic field at a distance r from cylinder axis for:– r < a (inside the wire)– r > a (outside the wire)
i
C
isdB 0
Current into page, circular
field lines
r
R
Ampere’s Law: Example 2 (cont)
C
isdB 0Current into
page, field tangent to the
closed amperian loop
r
enclosedirB 0)2(
2
22
22 )(
R
rir
R
irJienclosed
r
iB enclosed
20
20
2 R
irB
For r < R
Ampere’s Law: Example 2 (cont)
C
isdB 0Current into
page, field tangent to the
closed amperian loop enclosedirB 0)2(
0 0
2 2enclosedi i
Br r
0
2
iB
r
For r > R
r
20
2 R
irB
For r < R
B(r)
r
Solenoids
inBinhBhisdB
inhhLNiiNi
hBsdB
isdB
enc
henc
enc
000
0
)/(
000
Magnetic field of a magnetic dipoleA circular loop or a coil currying electrical current is a magnetic dipole, with magnetic dipole moment of magnitude m=NiA.Since the coil curries a current, it produces a magnetic field, that can be calculated using Biot-Savart’s law:
30
2/3220
2)(2)(
zzRzB
All loops in the figure have radius r or 2r. Which of these arrangements produce the largest magnetic field at the point indicated?
