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Physics 2102
Gabriela González
Physics 2102
Magnetic fields produced by
currents
?
The Biot-Savart Law
Jean-Baptiste
Biot (1774-1862)

dL
Felix Savart
(1791-1841)
i
• Quantitative rule for

r
computing the magnetic field
 
from any electric current

 0 idL  r
• Choose a differential element
dB 
of wire of length dL and
3
4 r
carrying a current i
• The field dB from this element 0 =4x10-7 T.m/A
at a point located by the vector (permeability constant)
r is given by the Biot-Savart
1 dq r
Law:
Compare with:
dE 
4 0 r
3
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 elements
 0 ids  r
dB 
4 r 3
 0 ids (r sin  )
dB 
4
r3
 0i  ds (r sin  )
B
4 
r3
Field of a straight wire
(continued)
sin   R / r
r  (s  R )
2
2 1/ 2


i
Rds
 0i ds (r sin  )
0

B

3

2
2 3/ 2
4

4 
r
  s  R 


 0i
Rds

2 0 s 2  R 2 3 / 2


 0i
0iR 
s


 2 2

1
/
2
2R
2  R s  R 2  
0
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?
 0i
B
2R
(4x107 T .m / A)(500 A)

2 (100m)
= 1 T!!
Recall that the earth’s magnetic
field is ~10-4T = 100 T
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?
 0 i ds  r
dB 
3
4 r
 0 idsR  0 iRd 
dB 

3
2
4 R
4 R
 0 id  0i 2  0i
B



4 R
4R
2R
ds
R
i
Direction of B?? Not
another right hand rule?!
Curl fingers around
direction of CURRENT.
Thumb points along
FIELD! Into page in this
case.
Forces between wires
Magnetic field due to wire 1
where the wire 2 is,
L I1
I2
F
0 I1
B1 
2 a
Force on wire 2 due to this field,
F21  L I 2 B1 
a
0 LI1 I 2
2 a
Ampere’s law:
Remember Gauss’ law?
Given an arbitrary closed surface, the electric flux through it is
proportional to the charge enclosed by the surface.
q
q
Flux=0!


Surface
  q
E  dA 
0
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”.
B

d
A

0

  q
E

d
A


0
Always! No isolated
magnetic charges
Ampere’s law:
a useful version of Gauss’ law.
 
 B  d s  0 I
i4
loop
The circulation of B
(the integral of B scalar
ds) along an imaginary
closed loop is proportional
to the net amount of current
traversing the loop.
i2
i3
i1
ds
 
 B  d s   0 (i1  i2  i3 )
loop
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: 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.
C
(Go around loop in same
direction as B field lines!)
R
 
 B  ds  0i
C
 Bds  B(2R)   i
0
 0i
B
2R
Ampere’s Law: Example
i
• 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)
r
Current into
page, circular
field lines
R
 
B

d
s


i
0

C
Ampere’s Law: Example 2 (cont)
 
B

d
s


i
0

C
r
Current into
page, field
tangent to the
closed
amperian loop
B(2r )  0ienclosed
 0ienclosed
B
2r
2
i
r
2
2
ienclosed  J (r )  2 r  i 2
R
R
 0ir
For r < R
B
2
2R
Ampere’s Law: Example 2 (cont)
 
B

d
s


i
0

r
Current into
page, field
tangent to the
closed
amperian loop
C
B(2r )  0ienclosed
0ienclosed 0i
B

2 r
2 r
0i
B
2 r
For r > R
 0ir
B
2R 2
For r < R
B(r)
r
Solenoids
 
 B  ds  0ienc
 
 B  ds  0  B h  0  0
ienc  iN h  i ( N / L)h  inh
 
 B  ds  0ienc  Bh  0inh  B  0in
Magnetic field of a magnetic dipole
A circular loop or a coil currying electrical current is a magnetic
dipole, with magnetic dipole moment of magnitude =NiA.
Since the coil curries a current, it produces a magnetic field, that can
be calculated using Biot-Savart’s law:



0
0 

B( z ) 

2
2 3/ 2
2 ( R  z )
2 z 3
All loops in the figure have radius r or 2r. Which of these
arrangements produce the largest magnetic field at the
point indicated?