Sources of the Sources of the Magnetic FieldMagnetic Field

March 23, 2009March 23, 2009

Note – These slides will be Note – These slides will be updated for the actual updated for the actual

presentation.presentation.

Remember the wire?

Try to remember…

30

20 4

1

4

1

r

dq

r

dq

rd

rrE

VECTOR UNITr

r

The “Coulomb’s Law” of Magnetism

A Vector Equation … duck

For the Magnetic Field,current “elements” create the field.

TmATm

typermeabilir

id

r

id unit

770

30

20

1026.1/104

44

:field fashion tosimilar aIn

rsrs

B

E

This is the Law ofBiot-Savart

Magnetic Field of a Straight Wire

•We intimated via magnets that the Magnetic field associated with a straight wire seemed to vary with 1/d.

•We can now PROVE this!

From the Past

Using a Compass

Right-hand rule: Grasp the element in your right hand with your extended thumb pointing in the direction of the current. Your fingers will then naturally curl around in the direction of the magnetic field lines due to that element.

Let’s Calculate the FIELD

Note:

For ALL current elementsin the wire:

ds X r

is into the page

The Details

02

0

20

)sin(

2B

it. DOUBLE and to0 from integrate

wesoamount equalan scontribute

wire theofportion Negative

)sin(

4

r

dsi

r

idsdB

Moving right along

R

i

Rs

RdsiB

SoRs

R

Rsr

22

)sin(sin

0

02/322

0

22

22

1/d

A bit more complicatedA finite wire

P1)sin()sin(: NOTE

r

ds

2/122

20 )sin(

4

)sin(

)sin(

Rsr

r

dsidB

r

R

rdsrds

More P1

R

i

whenRL

L

R

iB

and

Rs

dsiB

L

L

2B

,L 42

4

0

22

0

2/

2/2/322

0

P2

22

0

0

2/322

0

4

4

Rs

L

R

iB

or

Rs

dsiRB

L

APPLICATION: Find the magnetic field B at point P

Center of a Circular Arc of a Wire carrying current

More arc…

Cpoint at 4

44

44

0

0

0

02

0

20

20

R

iB

dR

i

R

iRddBB

R

iRd

R

idsdB

Rdds ds

The overall field from a circular current loop

Sorta looks like a magnet!

Iron

Howya Do Dat??

0rsdNo Field at C

Force Between Two Current Carrying Straight Parallel Conductors

Wire “a” createsa field at wire “b”

Current in wire “b” sees aforce because it is movingin the magnetic field of “a”.

The Calculation

d

iLi

iFd

iB

ba

b

a

2F

angles...right at are and Since

2

:calculatedjust what weis a"" wire

todue b"" at wire FIELD The

0

b""on

0b""at

BL

BL

Definition of the AmpereThe force acting between currents in parallel wires is the basis for the definition of the ampere, which is one of the seven SI base units. The definition, adopted in 1946, is this: The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length, of negligible circular cross section, and placed 1 m apart in vacuum, would produce on each of these conductors a force of magnitude 2 x 10-7 newton per meter of length.

Ampere’s Law

The return of Gauss

Remember GAUSS’S LAW??

0enclosedq

d AESurfaceIntegral

Gauss’s Law

•Made calculations easier than integration over a charge distribution.

•Applied to situations of HIGH SYMMETRY.

•Gaussian SURFACE had to be defined which was consistent with the geometry.

•AMPERE’S Law is the Gauss’ Law of Magnetism! (Sorry)

The next few slides have been lifted from Seb Oliveron the internet

Whoever he is!

Biot-Savart•The “Coulombs Law of Magnetism”

20 ˆ

4 r

rdsB

id

Invisible Summary

Biot-Savart Law (Field produced by wires)Centre of a wire loop radius RCentre of a tight Wire Coil with N turnsDistance a from long straight wire

Force between two wires Definition of Ampere

a

II

l

F

2

210

a

IB

2

0

R

IB

20

R

NIB

20

20 ˆ

4 r

rdsB

id

Magnetic Field from a long wirer

IB

2

0

cosdsBdsB

I

B

r

ds1cos0 dsBdsB

dsr

I

2

0 dsB

Using Biot-Savart Law

Take a short vector on a circle, ds

Thus the dot product of B & the short vector ds is:

Sum B.ds around a circular path

I

B

r

ds

dsr

I

2

0 dsB

dsB

Sum this around the whole ring

dsr

I

2

0 dsr

I

2

0

rds 2 Ιrr

I0

0 22

dsB

Circumference of circle

Consider a different path

•Field goes as 1/r

•Path goes as r.•Integral

independent of r

0 dsB

i

SO, AMPERE’S LAWby SUPERPOSITION:

We will do a LINE INTEGRATIONAround a closed path or LOOP.

Ampere’s Law

enclosedid 0 sB

USE THE RIGHT HAND RULE IN THESE CALCULATIONS

The Right Hand Rule .. AGAIN

Another Right Hand Rule

COMPARE

enclosedid 0 sB

0enclosedq

d AE

Line Integral

Surface Integral

Simple Example

Field Around a Long Straight Wire

enclosedid 0 sB

r

iB

irB

2

2

0

0

Field INSIDE a WireCarrying UNIFORM Current

The Calculation

rR

iB

andR

rii

irBdsBd

enclosed

enclosed

20

2

2

0

2

2

sB

R r

B

R

i

2

0

Procedure• Apply Ampere’s law only to highly

symmetrical situations.• Superposition works.

▫Two wires can be treated separately and the results added (VECTORIALLY!)

• The individual parts of the calculation can be handled (usually) without the use of vector calculations because of the symmetry.

• THIS IS SORT OF LIKE GAUSS’s LAW WITH AN ATTITUDE!

The figure below shows a cross section of an infinite conducting sheet carrying a current per unit x-length of l; the current emerges perpendicularly out of the page. (a) Use the Biot–Savart law and symmetry to show that for all points P above the sheet, and all points P´ below it, the magnetic field B is parallel to the sheet and directed as shown. (b) Use Ampere's law to find B at all points P and P´.

FIRST PART

Vertical ComponentsCancel

Apply Ampere to Circuit

Infinite Extent

B

B

Li

: thereforeis loop theinsideCurrent

lengthunit per current

L

The “Math”

Infinite Extent

B

B

20

0

0

B

LBLBL

id enclosedsB

Bds=0

A Physical Solenoid

Inside the Solenoid

For an “INFINITE” (long) solenoid the previous problem and SUPERPOSITION suggests that the field OUTSIDE this

solenoid is ZERO!

More on Long Solenoid

Field is ZERO!

Field is ZERO

Field looks UNIFORM

The real thing…..

Weak Field

Stronger - Leakage

Fairly Uniform field

Finite Length

Another Way

0

:

0

0

0

niB

nihBhh

id

Ampere

enclosed

sB

Application

•Creation of Uniform Magnetic Field Region•Minimal field outside

▫except at the ends!

Two Coils

“Real” Helmholtz Coils

Used for experiments.

Can be aligned to cancelout the Earth’s magneticfield for critical measurements.

The Toroid

Slightly lessdense than

inner portion

The Toroid

r

NiB

so

totalNirBd

Ampere

2

turns)# (N 2

:nintegratio ofpath the

in contained coil INNER about the

only worry need Weagain.

0

0

sB