PH212, APCS 2021-2022 3/24/2022

Chapter
Chapter30
29
Sources
Sourcesofofthe
theMagnetic
MagneticField
Field

Lake Tekapo, New Zealand

Magnetic Fields

The origin of the magnetic field is moving charges.

The magnetic field due to various current distributions can be calculated.
Ampère’s law is useful in calculating the magnetic field of a highly symmetric
configuration carrying a steady current.
Magnetic effects in matter can be explained on the basis of atomic magnetic
moments.

1
PH212, APCS 2021-2022 3/24/2022

Biot-Savart Law – Introduction

Biot and Savart conducted experiments on the force exerted by an electric

current on a nearby magnet.
They arrived at a mathematical expression that gives the magnetic field at
some point in space due to a current.
The magnetic field described by the Biot-Savart Law is the field due to a given
current-carrying conductor.
 Do not confuse this field with any external field applied to the conductor
from some other source.

Biot-Savart Law

Coulomb’s Law describes the E field due to infinitesimally

small charge

B field due to current in a short piece of wire is given by

the Biot-Savart Law
 μ I ds  ˆr
dB  o
4π r 2
• Strikingly similar except for:

Constant mo is the permeability of free space.

2
PH212, APCS 2021-2022 3/24/2022

Total Magnetic Field


dB is the field created by the current in the length segment ds.
• dB is perpendicular to both ds and r
• Size of dB depends on the angle between ds and r

To find the total field, sum up the contributions from all the current elements Ids
 μ I 
ds  ˆr
B o
4π  r2
 The integral is over the entire current distribution.
The law is also valid for a current consisting of charges flowing through
space.
 For example, this could apply to the beam in an accelerator.

Magnetic Field Compared to Electric Field

Distance
 The magnitude of the magnetic field varies as the inverse square of the
distance from the source.
 The electric field due to a point charge also varies as the inverse square of
the distance from the charge.
Direction
 The electric field created by a point charge is radial in direction.
 The magnetic field created
 by a current element is perpendicular to both
the length element ds and the unit vector r̂

3
PH212, APCS 2021-2022 3/24/2022

Magnetic Field Compared to Electric Field, cont.

Source
 An electric field is established by an isolated electric charge.
 The current element that produces a magnetic field must be part of an
extended current distribution.
 Therefore you must integrate over the entire current distribution.

 μ I 
Magnetic Field for a Long, Straight Conductor ds  ˆr
B o
4π  r2
The thin, straight wire is carrying a constant current

ds ˆr   dx cosθ  k
ˆ
Integrating over all the current elements gives 𝑥 = 𝑎 tan 𝜃
 μ I θ2
B  o  cos θ dθ kˆ
4πa θ1
μ I
 o  sin θ2  sin θ1  kˆ
4πa
q1 < 0, q2 > 0
If the conductor is an infinitely long, straight wire, q1 = -
-p/2 and q2 = p/2
 μ I
The field becomes B  o kˆ
2πa

4
PH212, APCS 2021-2022 3/24/2022

Magnetic Field for a Long, Straight Conductor

B field circles around (infinitely-long) current

• The field lines lie in planes perpendicular to the wire.
• Direction of B follows right-hand rule
• Magnitude B decreases as

cf. E field created by infinitely-long linear charge

distribution was

• Similar, but E points outward, B rotates around

Magnetic Field of a Wire

Here 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.
If the current is reversed, the direction of the
needles also reverse.

by the iron filings

5
PH212, APCS 2021-2022 3/24/2022

Magnetic Field for a Curved Wire Segment

Find the field at point O due to the wire segment.

Integrate, remembering I and R are constants

μo I
B θ 𝑑𝑠⃗ × 𝑟̂ 𝑑𝑠 𝑘
4πa =
𝑟 𝑎
 q will be in radians
 The field points into the page
Magnetic Field for a Circular Loop of wire
 θ = 2π
μo I μ I μ I
B θ  o 2π  o
4πa 4πa 2a
This is the field at the center of the loop.

Magnetic Field for a Circular Current Loop

The loop has a radius of R and carries a

steady current of I.

μo I a2
Bx 
 
3
2 a2  x 2 2

Consider the field at the center of the current

loop (x = 0)
μo I a 2 μo I
Bx  
 
3
2 a2  x 2 2 2a

6
PH212, APCS 2021-2022 3/24/2022

Magnetic Field Lines for a Loop

B field at the center of of the loop (x = 0)
B field is axially symmetric
• The lines (left figure) are drawn for the plane that contains the axis of the
loop
• Look like the pattern around a bar magnet (right figure)
 We can define N & S poles for a loop

Magnetic Force Between Two Parallel Conductors

Two parallel wires each carry a steady current.

The magnetic force between the two wires :

μo I 1 I 2
F1  
2πa 𝐹⃗ = 𝐼 𝑙⃗ × 𝐵

 Parallel conductors carrying currents in the

same direction attract each other.
 Parallel conductors carrying current in
opposite directions repel each other.

μo I 2
B2 
2πa

7
PH212, APCS 2021-2022 3/24/2022

Magnetic Force Between Two Parallel Conductors, final

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

FB μo I1 I 2

 2πa
The derivation assumes both wires are long compared with their separation
distance.
 Only one wire needs to be long.
 The equations accurately describe the forces exerted on each other by a
long wire and a straight, parallel wire of limited length, ℓ.

Definition of the Ampere and the Coulomb

The force between two parallel wires can be used to define the ampere.
For I1 = I2 = 1A and a = l = 1m

 𝜇 = 4𝜋 × 10 𝑁/𝐴 exact

The SI unit of charge, the coulomb, is defined in terms of the ampere.

𝑞 1𝐶
𝐼= → 1𝐴 =
𝑡 1𝑠

8
PH212, APCS 2021-2022 3/24/2022

André-Marie Ampère

1775 – 1836
French physicist
Credited with the discovery of
electromagnetism
 The relationship between electric
current and magnetic fields
Also worked in mathematics

Magnetic Field for a Long, Straight Conductor: Direction

 
The product of B  ds can be evaluated
for small length elements ds on the
circular path defined by the compass
needles for the long straight wire.

𝜇 𝐼
𝐵 ∙ 𝑑𝑠⃗ = 𝐵𝑑𝑠 = 𝑑𝑠
2𝜋𝑎
𝜇 𝐼
𝐵 ∙ 𝑑𝑠⃗ = 𝑑𝑠 = 𝜇 𝐼
2𝜋𝑎

μo I
B
2πa

9
PH212, APCS 2021-2022 3/24/2022

Ampere’s Law
 
Ampere’s law states that the line integral of B  ds around any closed path
equals moI where I is the total steady current passing through any surface
bounded by the closed path:
 
  ds  μo I
B

Put the thumb of your right hand in the direction of the positive current
through the amperian loop and your fingers curl in the direction you should
integrate around the loop.
Ampère’s Law follows Biot-Savart Law
• Same way as Gauss’s Law follows Coulomb’s Law
• It’s useful when a symmetry of the problem helps us to predict at least the
direction of the B field

Boundary

- What is the boundary of an open path?

- What is the boundary of an open surface?
- What is the boundary of a volume?
- What is the boundary of a closed surface?
- What is the boundary of a closed path ?

10
PH212, APCS 2021-2022 3/24/2022

Field Due to a Long Wire with Finite Thickness

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  use Ampère’s Law
 For r ≥ R, the same result as Biot-Savart Law.

 For r < R, we need I’, the current inside the

amperian circle.

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.

11
PH212, APCS 2021-2022 3/24/2022

Magnetic Field of a Solenoid

A solenoid is a long wire wound in the form of a

helix.
The field lines in the interior are
 Nearly parallel to each other
 Uniformly distributed
 Close together
This indicates the field is strong and almost
uniform.

Magnetic Field of a Tightly Wound Solenoid

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.

12
PH212, APCS 2021-2022 3/24/2022

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
Applying Ampere’s Law gives to loop 2
   
 B  ds  
path 1
B  ds  B 
path 1
ds  B

The total current through the rectangular path equals the

current through each turn multiplied by the number of turns.
 
 B  ds  B  m NI o B  μo
N

I  μo n I

 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.
.

Magnetic Flux

The magnetic flux associated with a magnetic field

is defined in a way similar to electric flux.
Consider an area element dA on an arbitrarily
shaped surface.

The magnetic field in this element is B.

dA is a vector that is perpendicular to the surface
and has a magnitude equal to the area dA.
The magnetic flux ΦB is
 
 B   B  dA
The unit of magnetic flux is T.m2 = Wb
 Wb is a weber

13
PH212, APCS 2021-2022 3/24/2022

Magnetic Flux Through a Planes


A special case is when a plane of area A
makes an angle θ with dA.
The magnetic flux is ΦB = BA cos θ.
In this case, the field is parallel to the
plane and ΦB = 0.

In this case, the field is perpendicular

to the plane and  = BA.
 This is the maximum value of the
flux.

Gauss’ Law in Magnetism

Magnetic fields do not begin or end at any point.

 Magnetic field lines are continuous and form closed loops.
 The number of lines entering a closed surface equals the number of
lines leaving the surface.
Gauss’ law in magnetism says the magnetic flux through any closed surface
is always zero:
 
 B  dA  0
This indicates that isolated magnetic poles (monopoles) have never been
detected.
 Perhaps they do not exist
 Certain theories do suggest the possible existence of magnetic monopoles.

14
PH212, APCS 2021-2022 3/24/2022

Magnetism in Matter

Different materials react differently to magnetic field

• Some (e.g. iron) stick and others don’t
• Read textbook section 30.6 (9th ed)
Three types of magnetism
• Ferromagnetism – strongly attracted to magnets
 Iron (ferrum), cobalt, nickel, etc.
• Paramagnetism – weakly attracted to magnets
 Ferromagnetic metals turn paramagnetic at high temperature
• Diamagnetism – weakly repelled by magnets
 Majority of materials
 Special case: superconductors are strongly diamagnetic

Magnetic Moments
In general, any current loop has a magnetic field and thus has a magnetic dipole
moment.
This includes atomic-level current loops described in some models of the atom.
This will help explain why some materials exhibit strong magnetic properties.

15
PH212, APCS 2021-2022 3/24/2022

Magnetic Moments – Classical 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.

L is the angular momentum.

m is magnetic moment.

Magnetic Moments – Classical Atom, cont.

This model assumes the electron moves:
 with constant speed v
 in a circular orbit of radius r
 travels a distance 2pr in a time interval T
The current associated with this orbiting electron is

e ev
I 
T 2πr
1
The magnetic moment is μ  I A  evr
2
The magnetic moment can also be expressed in terms of the angular momentum.
 e 
μ L
 2me 

16
PH212, APCS 2021-2022 3/24/2022

Magnetic Moments – Classical Atom, final

The magnetic moment of the electron is proportional to its orbital angular

momentum.
 
 The vectors L and m point in opposite directions.
 Because the electron is negatively charged
Quantum physics indicates that angular momentum is quantized.

Magnetic Moments of Multiple Electrons

In most substances, the magnetic moment of one electron is canceled by that of

another electron orbiting in the same direction.
The net result is that the magnetic effect produced by the orbital motion of the
electrons is either zero or very small.

17
PH212, APCS 2021-2022 3/24/2022

Electron Spin

Electrons (and other particles) have an intrinsic property called spin that also
contributes to their 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

Electron Spin, cont.

The classical model of electron spin is the

electron spinning on its axis.
The magnitude of the spin angular
momentum is
3
S 
2
  is Planck’s constant.

18
PH212, APCS 2021-2022 3/24/2022

Electron Spin and Magnetic Moment

The magnetic moment characteristically associated with the spin of an electron
has the value
e
μspin 
2me
This combination of constants is called the Bohr magneton mB = 9.27 x 10-24 J/T.

Electron Magnetic Moment, final

The total magnetic moment of an atom

is the vector sum of the orbital and spin
magnetic moments.
Some examples are given in the table
at right.
The magnetic moment of a proton or
neutron is much smaller than that of an
electron and can usually be neglected.

19
PH212, APCS 2021-2022 3/24/2022

Ferromagnetism

Some substances exhibit strong magnetic effects called ferromagnetism.

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.

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.

20
PH212, APCS 2021-2022 3/24/2022

Domains, Unmagnetized Material

The magnetic moments in the domains

are randomly aligned.
The net magnetic moment is zero.

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.

21
PH212, APCS 2021-2022 3/24/2022

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 a net magnetization
in the direction of the original field.

Curie Temperature

The Curie temperature is the critical temperature above which a ferromagnetic

material loses its residual magnetism.
 The material will become paramagnetic.
Above the Curie temperature, the thermal agitation is great enough to cause a
random orientation of the moments.

22
PH212, APCS 2021-2022 3/24/2022

Paramagnetism
Paramagnetic substances have small but positive magnetism.
It results from the presence of atoms that have permanent magnetic moments.
 These moments interact weakly with each other.
When placed in an external magnetic field, its atomic moments tend to line up
with the field.
 The alignment process competes with thermal motion which randomizes the
moment orientations.

Diamagnetism

When an external magnetic field is applied to a diamagnetic substance, a weak

magnetic moment is induced in the direction opposite the applied field.
Diamagnetic substances are weakly repelled by a magnet.
 Weak, so only present when ferromagnetism or paramagnetism do not exist

23
PH212, APCS 2021-2022 3/24/2022

Meissner Effect

Certain types of superconductors also

exhibit perfect diamagnetism in the
superconducting state.
 This is called the Meissner effect.
If a permanent magnet is brought near
a superconductor, the two objects repel
each other.

24