Physics 2102 Jonathan Dowling Physics 2102 Lecture 15

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.