Physics 2102Physics 2102Lecture 15Lecture 15
Biot-Savart Biot-Savart LawLaw
Physics 2102
Jonathan Dowling
Jean-Baptiste Biot(1774-1862)
Felix Savart (1791–1841)
What Are We Going to Learn?What Are We Going to Learn?A Road MapA Road Map
• Electric charge Electric force on other electric charges Electric field, and electric potential
• Moving electric charges : current• Electronic circuit components: batteries, resistors, capacitors• Electric currents Magnetic field
Magnetic force on moving charges• Time-varying magnetic field Electric Field• More circuit components: inductors.• Electromagnetic waves light waves• Geometrical Optics (light rays).• Physical optics (light waves)
Electric Current:Electric Current:A Source of Magnetic FieldA Source of Magnetic Field
IB
Wire withcurrentINTO page
B
• Observation: anelectric currentcreates a magneticfield
• Simple experiment:hold a current-carrying wire near acompass needle!
Yet Another Right Hand Rule!Yet Another Right Hand Rule!• Point your thumb along the direction of the
current in a straight wire
• The magnetic field created by the current
consists of circular loops directed along your
curled fingers.
• The magnetic field gets weaker with distance.
• You can apply this to ANY straight wire (even
a small differential element!)
• What if you have a curved wire? Break into
small elements.
iB
Direction of B??? i
SuperpositionSuperposition• Magnetic fields (like electric
fields) can be “superimposed” --just do a vector sum of B fromdifferent sources
• The figure shows four wireslocated at the 4 corners of asquare. They carry equalcurrents in directions indicated
• What is the direction of B at thecenter of the square?
B
OUT OUT
IN IN
When we computed the electric field due to charges we usedCoulomb’s law. If one had a large irregular object, one broke itinto infinitesimal pieces and computed,
rr
dqEd ˆ
4
1
2
0!"
=r Which we write as,
3
04 r
rdqEd
rr
!"=
If you wish to compute the magnetic field due to acurrent in a wire, you use the law of Biot and Savart.
Biot-SavartBiot-Savart Law Law
The The Biot-Savart Biot-Savart LawLaw
• Quantitative rule forcomputing the magnetic fieldfrom any electric current
• Choose a differential elementof wire of length dL andcarrying a current i
• The field dB from this elementat a point located by the vectorr is given by the Biot-SavartLaw
Ldr
rr
3
0
4 r
rLidBd
rrr !=
"
µ
i
µ0 =4πx10-7 Tm/A(permeability constant)
Jean-Baptiste Biot (1774-1862)
Felix Savart (1791-1841)
3
0
Compare with 4
dq rdE
r!"=
rr
Biot-Savart Biot-Savart LawLaw
• An infinitely long straight wirecarries a current i.
• Determine the magnetic fieldgenerated at a point located at aperpendicular distance R fromthe wire.
• Choose an element ds as shown
• Biot-Savart Law: dB pointsINTO the page
• Integrate over all such elements
3
0
4 r
rsidBd
rrr !=
"
µ
3
0 )sin(
4 r
ridsdB
!
"
µ=
!"
"#
=3
0 )sin(
4 r
rdsiB
$
%
µ
( )!"
"# +=
2/322
0
4 Rs
Rdsi
$
µ
( )!"
+=
0
2/322
0
2 Rs
Rdsi
#
µ
( )
!
""#
$
%%&
'
+=
0
2/1222
0
2 RsR
siR
(
µ
R
i
!µ
2
0=
rR /sin =!2/122 )( Rsr +=
!"
"#
=3
0 )sin(
4 r
rdsiB
$
%
µ
Field of a Straight WireField of a Straight Wire
3
0
4 r
rsidBd
rrr !=
"
µ3
0 )sin(
4 r
ridsdB
!
"
µ=
Example : A Practical MatterExample : A Practical Matter
A power line carries acurrent of 500 A. What isthe magnetic field in a houselocated 100 m away fromthe power line?
R
iB
!
µ
2
0=
)100(2
)500)(/.104( 7
m
AAmTx
!
!"
=
= 1 µT!!
Recall that the earth’s magneticfield is ~10–4T = 100 µT
Biot-Savart Biot-Savart LawLaw• A circular arc of wire of radius
R carries a current i.• What is the magnetic field at
the center of the loop?
3
0
4 r
rdsiBd
rr !=
"
µ
2
0
3
0
44 R
iRd
R
idsRdB
!
"
µ
"
µ==
0 0
4 4
iidB
R R
µ µ!" "
#= =$
Direction of B?? Not anotherright hand rule?!TWO right hand rules!:If your thumb points along theCURRENT, your fingers willpoint in the same direction as theFIELD.If you curl our fingers arounddirection of CURRENT, yourthumb points along FIELD!
i
a
IB
!
µ
2
10
1=
Magnetic field due to wire 1 where the wire 2 is,
1221BILF =
a
I2I1L
a
IIL
!µ
2
210=
Force on wire 2 due to this field,F
Forces between wiresForces between wires
SummarySummary• Magnetic fields exert forces on moving charges:
• The force is perpendicular to the field and the velocity.• A current loop is a magnetic dipole moment.• Uniform magnetic fields exert torques on dipole moments.
• Electric currents produce magnetic fields:•To compute magnetic fields produced by currents, use Biot-Savart’s law for each element of current, and then integrate.• Straight currents produce circular magnetic field lines, withamplitude B=µ0i/2πr (use right hand rule for direction).• Circular currents produce a magnetic field at the center(given by another right hand rule) equal to B=µ0iΦ/4πr
• Wires currying currents produce forces on each other:parallel currents attract, antiparallel currents repel.
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