Lecture 22:Lecture 22: MON 09 MARMON 09 MARCh.29.1Ch.29.1––2: Magnetic Fields Due to Currents2: Magnetic Fields Due to Currents

Biot-Savart Biot-Savart LawLaw

Physics 2102

Jonathan Dowling

Jean-BaptisteBiot (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: an electriccurrent creates amagnetic field

• Simple experiment: holda current-carrying wirenear a compass needle!

B

Hans Christian Oersted was a professor ofscience at Copenhagen University. In 1820he arranged in his home a sciencedemonstration to friends and students. Heplanned to demonstrate the heating of awire by an electric current, and also tocarry out demonstrations of magnetism, forwhich he provided a compass needlemounted on a wooden stand.

While performing his electricdemonstration, Oersted noted to hissurprise that every time the electriccurrent was switched on, the compassneedle moved. He kept quiet and finishedthe demonstrations, but in the months thatfollowed worked hard trying to make senseout of the new phenomenon.

CaseyCasey’’s Right Hand Rule!s Right Hand Rule!• Point your thumb along the direction of

the current in a straight wire

• The magnetic field created by thecurrent consists of circular loops directedalong your curled fingers.

• The magnetic field gets weaker withdistance: For long wire it’s a 1/R Law!

• You can apply this to ANY straight wire(even a small differential element!)

• What if you have a curved wire? Breakinto small elements.

iB

Direction of B! i

SuperpositionSuperposition• Magnetic fields (like electric

fields) can be “superimposed”-- just do a vector sum of Bfrom different sources

• The figure shows four wireslocated at the 4 corners ofa square. They carry equalcurrents in directionsindicated

• What is the direction of Bat the center of the square?

B

I-OUT I-OUT

I-IN I-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!"

=!

Which we write as,

If you wish to compute the magnetic field due to acurrent in a wire, you use the law of Biot and Savart.

Biot-Savart Biot-Savart LawLaw

The The Biot-Savart Biot-Savart LawLaw

• Quantitative rule for computingthe magnetic field from anyelectric current

• Choose a differential elementof wire of length dL andcarrying a current i

• The field dB from this elementat a point located by thevector r is given by the Biot-Savart Law

3

0

4 r

rLidBd

!!! !=

"

µ

µ0 =4πx10–7 T•m/A(permeability constant)

Jean-Baptiste Biot (1774-1862)

Felix Savart (1791-1841)Ld

!

r!

i

dB

!

d! E =

1

4"#0

dq

r2

" r

Ld!

r!

i

dB

!

d! B =

µ0

4"

id! L #" r

r2

Biot-Savart Law for B-Fields

Coulomb Law for E-Fields

Biot-Savart Requires ARight-Hand Rule

Both Are 1/r2 Laws!The has no units.

!

! r

Biot-Savart Biot-Savart LawLaw

• An infinitely long straight wirecarries a current i.

• Determine the magnetic fieldgenerated at a point located ata perpendicular distance R fromthe wire.

• Choose an element ds as shown• Biot-Savart Law: dB points

INTO the page• Integrate over all such

elements

3

0

4 r

rsidBd

!!! !=

"

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

!!! !=

"

µ3

0 )sin(

4 r

ridsdB

!

"

µ=

Is the B-Field From a Power Line Dangerous?Is the B-Field From a Power Line Dangerous?

A power linecarries a currentof 500 A.

What is themagnetic field in ahouse located100 m away fromthe power line?

R

iB

!µ

2

0=

!

=(4" #10$7T %m /A)(500A)

2" (100m)

= 1 µT

Recall that the earth’s magneticfield is ~10–4T = 100 µT

Probably not dangerous!

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

!! !=

"

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 will pointin the same direction as theFIELD.

If you curl our fingers arounddirection of CURRENT, yourthumb points along FIELD!

i