1 22.7 Source of magnetic field due to current The Oersted’s discovery in 1819 indicates an electric current can act as a source of magnetic field. Biot.

1

22.7 Source of magnetic field due to current

The Oersted’s discovery in 1819 indicates an electric current can act as a source of magnetic field.

Biot and Savart investigated the force exerted by an electric current on a nearby magnet in the 19th century.

They arrived at a mathematical expression for the magnetic field at some point in space due to an electric current.

2

Biot-Savart Law The magnetic field is at some

point P An infinitesimal length element is The wire is carrying a steady

current of I The vector is perpendicular to

both ds and the unit vector directed from the element toward P

The magnitude of is inversely proportional to r2, where r is the distance from the element to P

dB

ds

dB

r̂

dB

3

Observations The magnitude of is proportional to

the current and to the magnitude ds of the length element ds

The magnitude of is proportional to sin where is the angle between the vectors and r̂

dB

dB

ds

4

The observations are summarized in the mathematical equation called Biot-Savart Law:

The Biot-Savart law gives the magnetic field only for a small length of the conductor

Biot-Savart Law, Equation

2

ˆm

I dd k

r

s rB

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Permeability of Free Space

The constant o is called the permeability of free space

o = 4 x 10-7 T. m / A The Biot-Savart Law can be written as

7104

om

T mk

A

2

ˆ

4o I d

dr

s rB

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Total Magnetic Field due a wire of current

To find the total field, you need to sum up the contributions from all the current elements You need to evaluate the field by

integrating over the entire current distribution

Exercise 52

7

The magnetic field at the point P is

01 2(cos cos )

4

IB

a

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B for an Infinite-Long, Straight Conductor, Direction

The magnetic field lines are circles concentric with the wire

The field lines lie in planes perpendicular to to wire

The magnitude of the field is constant on any circle of radius a

The right hand rule for determining the direction of the field is shown 2

oIBr

9

10

11

12

B for a Circular Current Loop The loop has a radius of

R and carries a steady current of I

Find at point P

The field at the center of the loop

2

32 2 22

ox

IRB

x R

B

R

IBx 2

0

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22.8 Magnetic Force Between Two Parallel Conductors

Two parallel wires each carry a steady current

The field due to the current in wire 2 exerts a force on wire 1 of F1 = I1l B2

Substituting the equation for B2 gives

Parallel conductors carrying currents in the same direction attract each other

Parallel conductors carrying current in opposite directions repel each other

2B

1 21 2

oI IF

a

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Magnetic Force Between Two Parallel Conductors

The result is often expressed as the magnetic force between the two wires, FB

This can also be given as the force per unit length,

a2IIF 21oB

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22.9 Definition of the Ampere The force between two parallel wires

can be used to define the ampere When the magnitude of the force per

unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

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Magnetic Field of a Wire Without Current

A compass can be used to detect the magnetic field

When there is no current in the wire, there is no field due to the current

The compass needles all point toward the earth’s north pole

Due to the earth’s magnetic field

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Magnetic Field of a Wire With a Current

The wire carries a strong current

The compass needles deflect in a direction tangent to the circle

This shows the direction of the magnetic field produced by the wire

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André-Marie Ampère 1775 –1836 Credited with the

discovery of electromagnetism The relationship

between electric currents and magnetic fields

Died of pneumonia

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Ampere’s Law The product of can be evaluated for

small length elements on the circular path defined by the compass needles for the long straight wire

Ampere’s Law states that the line integral of

around any closed path equals oI where I is the total steady current passing through any surface bounded by the closed path

od I B s

dB s

ds

dB s

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Field Due to a Long Straight Wire with a finite radius

Want to calculate the magnetic field at a distance r from the center of a wire carrying a steady current I

The current is uniformly distributed through the cross section of the wire

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Field Due to a Long Straight Wire - From Ampere’s law

Outside of the wire, r > R

Inside the wire, we need I’, the current inside the amperian circle

(2 )

2

o

o

d B r I

IB

r

B s

2

2

2

(2 ) ' '

2

o

o

rd B r I I I

RI

B rR

B s

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Field Due to a Long Straight Wire – Summary

The field is proportional to r inside the wire

The field varies as 1/r outside the wire

Both equations are equal at r = R

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Magnetic Field of a Toroid Find the field at a

point at distance r from the center of the toroid

The toroid has N turns of wire

(2 )

2

o

o

d B r NI

NIB

r

B s

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22.10 Magnetic Field of a Solenoid A solenoid is a long wire wound in the form

of a helix A reasonably uniform magnetic field can be

produced in the space surrounded by the turns of the wire

Each of the turns can be modeled as a circular loop The net magnetic field is the vector sum of all the

fields due to all the turns

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Ideal Solenoid – Characteristics An ideal solenoid is

approached when The turns are closely

spaced The length is much

greater than the radius of the turns

For an ideal solenoid, the field outside of solenoid is negligible

The field inside is uniform

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Ampere’s Law Applied to a Solenoid, cont Applying Ampere’s Law gives

The total current through the rectangular path equals the current through each turn multiplied by the number of turns

1 1path path

d d B ds B B s B s

od B NI B s

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Magnetic Field of a Solenoid, final

Solving Ampere’s Law for the magnetic field is

n = N / l is the number of turns per unit length

This is valid only at points near the center of a very long solenoid

o o

NB I nI

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Magnetic Field of a Solenoid with a finite length

The field distribution is similar to that of a bar magnet

As the length of the solenoid increases

The interior field becomes more uniform The exterior field becomes weaker

The field lines in the interior are Approximately parallel to each other Uniformly distributed Close together

This indicates the field is strong and almost uniform

Magnetic field lines of a solenoid

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22.11 Magnetic Moment – Bohr Atom

The electrons move in circular orbits

The orbiting electron constitutes a tiny current loop

The magnetic moment of the electron is associated with this orbital motion

The angular momentum and magnetic moment are in opposite directions due to the electron’s negative charge

31

Magnetic Moments of Multiple Electrons

In most substances, the magnetic moment of one electron is canceled by that of another electron orbiting in the opposite direction

The net result is that the magnetic effect produced by the orbital motion of the electrons is either zero or very small

32

Electron Spin Electrons (and other particles) have an

intrinsic property called spin that also contributes to its magnetic moment The electron is not physically spinning It has an intrinsic angular momentum as if

it were spinning Spin angular momentum is actually a

relativistic effect

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Electron Magnetic Moment In atoms with multiple

electrons, many electrons are paired up with their spins in opposite directions The spin magnetic

moments cancel Those with an “odd” electron

will have a net moment Some moments are given in

the table

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Ferromagnetic Materials Some examples of ferromagnetic materials are

Iron Cobalt Nickel Gadolinium Dysprosium

They contain permanent atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field

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Domains All ferromagnetic materials are made up

of microscopic regions called domains The domain is an area within which all

magnetic moments are aligned The boundaries between various

domains having different orientations are called domain walls

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Domains, Unmagnetized Material

The magnetic moments in the domains are randomly aligned

The net magnetic moment is zero

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Domains, External Field Applied

A sample is placed in an external magnetic field

The size of the domains with magnetic moments aligned with the field grows

The sample is magnetized

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Domains, External Field Applied, cont

The material is placed in a stronger field

The domains not aligned with the field become very small

When the external field is removed, the material may retain most of its magnetism

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22.12 Magnetic Levitation The Electromagnetic System (EMS) is

one design model for magnetic levitation

The magnets supporting the vehicle are located below the track because the attractive force between these magnets and those in the track lift the vehicle

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EMS

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German Transrapid – EMS Example

Close surface and line integrals of Electric and Magnetic fields

42

Electric field, Magnetic field,

Close Surface Integral

Close Line Integral

0 in

S

qAdE

E

tot

C

IμldB 0

(Gauss Law)

(Ampere’s Law)

？

？

B

Line integral of electric field along a close path

43

ldE VVb

aab

Potential difference between a and b points,

For a close path, a=b and Va=Vb.

0 ldEc

The electric force is conservative.

44

Magnetic Flux The magnetic flux

is expressed as The flux depends on

the magnetic field and the area:

B d B A

Surface integral of magnetic field around any close surfaces

45

0S

AdB

No magnetic monopole is found so far!

Close surface and line integrals of Electric and Magnetic fields

46

Electric field, Magnetic field,

Close Surface Integral

Close Line Integral

0 in

S

qAdE

E

tot

C

IμldB 0

(Gauss Law)

(Ampere’s Law)

B

0S

AdB

No magnetic monopole!

0 ldEc

The electric force is conservative.

Steady state only

47

Exercises of Chapter 22 5, 8, 14, 18, 20, 22, 23, 29, 35, 38, 44,

49, 55, 68, 69