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 rrLidBd

i

0 =4x10-7 T.m/A(permeability constant)

Jean-Baptiste Biot (1774-1862)

?Felix Savart (1791-1841)

Compare with: 30

14

dq rdEr

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 rrsidBd

30 )sin(

4 rridsdB

30 )sin(

4 rrdsiB

2/322

0

4 RsRdsi

02/322

0

2 RsRdsi

0

2/12220

2 RsRsiR

Ri

2

0

rR /sin

2/122 )( Rsr

30 )sin(

4 rrdsiB

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?

RiB

2

0

)100(2)500)(/.104( 7

mAAmTx

= 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?

dsR

30

4 rrdsiBd

20

30

44 RiRd

RidsRdB

Ri

Ri

RidB

242

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

aIB

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

0qAdE

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(

RiB

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 linesr

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 )(

Rrir

RirJienclosed

r

iB enclosed

20

20

2 RirB

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

2iBr

For r > R

r

20

2 RirB

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 =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?