26. Magnetism: Force & Field

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26. Magnetism: Force &amp; Field. Topics. The Magnetic Field and Force The Hall Effect Motion of Charged Particles Origin of the Magnetic Field Laws for Magnetism Magnetic Dipoles Magnetism. Introduction. An electric field is a disturbance in space caused

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### 26. Magnetism: Force & Field

Topics
• The Magnetic Field and Force
• The Hall Effect
• Motion of Charged Particles
• Origin of the Magnetic Field
• Laws for Magnetism
• Magnetic Dipoles
• Magnetism
Introduction

An electric field is a disturbance in space caused

by electric charge. A magnetic field is a

disturbance in space caused by moving electric

charge.

An electric field creates a force on electric charges.

A magnetic field creates a force on moving electric

charges.

Magnetic Field and Force

It has been found that the

magnetic force depends on

the angle between the velocity

of the electric charge and

the magnetic field

Magnetic Field and Force

The force on a moving charge can

be written as

where B

represents the

magnetic field

Magnetic Field and Force

The SI unit of magnetic field is the tesla

(T) = 1 N /(A.m). But often we use a smaller

unit: the gauss (G) 1 G = 10-4 T

### The Hall Effect

h

The Hall Effect

Consider a magnetic field into the page and a current

flowing from left to right.

Free positive

charges will be

deflected upwards

and free negative

charges

downwards.

h

The Hall Effect

Eventually, the induced electric force balances the

magnetic force:

Hall

Voltage

t is the thickness

Hall coefficient

### Motion of Charged Particles in a Magnetic Field

Motion of Charged Particles in a Magnetic Field

The magnetic force on

a point charge

does no work. Why?

The force merely changes

the direction of motion of

the point charge.

Motion of Charged Particles in a Magnetic Field

Newton’s 2nd Law

Motion of Charged Particles in a Magnetic Field

Since,

the cyclotron period is

Its inverse is the cyclotron frequency

### Origin of the Magnetic Field

The Biot-Savart Law

A point charge produces an electric field.

When the charge moves it produces a

magnetic field, B:

m0 is the magnetic

constant:

As drawn, the field

is into the page

The Biot-Savart Law

When the expression for B is extended

to a current element, IdL,

we get the Biot-Savart law:

The total field is found by

integration:

P

Biot-Savart Law: Example

The magnetic field due to an infinitely long current

can be computed from the Biot-Savart law:

x

Biot-Savart Law: Example

Note: if your right-hand thumb points in the

direction of the current, your fingers will curl in the

direction of the resulting

magnetic field

I

### Laws of Magnetism

Magnetic Flux

Just as we did for electric fields, we

can define a flux for a magnetic

field:

But there is a profound difference

between the two kinds of flux…

Gauss’s Law for Magnetism

Isolated positive and negative electric

charges exist. However, no one has ever

found an isolated magnetic north or south

pole, that is, no one has ever found a

magnetic monopole

Consequently, for any closed surface the

magnetic flux into the surface is exactly

equal to the flux out of the closed surface

Gauss’s Law for Magnetism

This yields Gauss’s law for magnetism

Unfortunately, however, because this law

does not relate the magnetic field to its

source it is not useful for computing

magnetic fields. But there is a law that is…

I

Ampere’s Law

If one sums the dot product around

a closed loop that encircles a steady current

I then Ampere’s law holds:

That law can be used to compute magnetic fields, given a problem of sufficient symmetry

z

y

x

Ampere’s Law: Example

What’s the magnetic field a distance z above an

infinite current sheet of current density l per unit

length in the y direction? From symmetry, the magnetic

field must point in the

positive y direction

above the sheet and in

the negative y direction

below the sheet.

z

y

x

Ampere’s Law: Example

Ampere’s law states that the line integral of the

magnetic field along any closed loop is equal to m0

times the current it encircles:

Draw a rectangular

loop of height

2a in z and length b

in y, symmetrically

sheet.

z

y

x

Ampere’s Law: Example

The only contribution to the integral is from the upper

and lower segments of the loop. From symmetry the

magnitude of the magnetic field is constant and the

same on both segments. Therefore,

the integral is just 2Bb.

The encircled current is

I = l b. So, Ampere’s

law gives 2Bb = m0l b and

therefore B = m0 l / 2

### Magnetic Force on a Current

Magnetic Force on a Current

Force on each charge:

Force on wire segment:

n = number of charges

per unit volume

Magnetic Force on a Current

Note the direction

of the force on

the wire

For a current element

IdL the force is

Magnetic Force Between Conductors

Since the force on a current-carrying

wire in a magnetic field is

two parallel wires,

with currents I1 and I2 exert

a magnetic force on each

other. The force on wire 2 is:

d

### Magnetic Dipoles

Magnetic Moment

A current loop experiences no net force

in a uniform magnetic field. But it does

experience a

F torque

B

The force is

F = IaB

F

Magnetic Moment

Magnitude of torque

where A= ab

For a loop with N turns, the

torque is

Magnetic Moment

It is useful to define a new vector

quantity called the magnetic dipole

moment

then we can write the torque as

Magnetic Moment

The magnetic torque that causes the

dipole to rotate does work and tends to

decrease the potential energy of the

magnetic dipole

If we agree to set the potential energy to zero

at 90o then the potential energy is given by

### Magnetization

Magnetization

Atoms have magnetic dipole moments due to

• orbital motion of the electrons
• magnetic moment of the electron

When the magnetic

moments align we

say that the material

is magnetized.

Types of Materials

Materials exhibit three types of magnetism:

• paramagnetic
• diamagnetic
• ferromagnetic
Paramagnetism

Paramagnetic materials

• have permanent magnetic moments
• moments randomly oriented at normal temperatures
Paramagnetism
• Small effect (changes B by only 0.01%)
• Example materials
• Oxygen, aluminum, tungsten, platinum
Diamagnetism

Diamagnetic materials

• no permanent magnetic moments
• magnetic moments induced by applied magnetic field B
• applied field creates magnetic moments opposed to the field
Diamagnetism

Common to all materials.

Applied B field induces a magnetic field opposite the applied field, thereby weakening the overall magnetic field

But the effect is very small: Bm ≈ -10-4 Bapp

Diamagnetism

Example materials

• high temperaturesuperconductors
• copper
• silver
Ferromagnetism

Ferromagnetic materials

• have permanent magnetic moments
• align at normal temperatures when an external field is applied and strongly enhances applied magnetic field
Ferromagnetism

Ferromagnetic materials

(e.g. Fe, Ni, Co, alloys)

have domains of randomly

aligned magnetization

(due to strong interaction

of magnetic moments of neighboring

atoms)

Ferromagnetism

Applying a magnetic field causes domains

aligned with the applied field to grow at

the expense of others that shrink

Saturation magnetization is reached

when the aligned domains

have replaced all others

Ferromagnetism

In ferromagnets, some magnetization

will remain after the applied

field is reduced to zero,

yielding permanent magnets

Such materials exhibit

hysteresis

Summary
• Magnetic Force
• Perpendicular to velocity and field
• Does no work
• Changes direction of motion of charged particle
• Motion of Point Charge
Summary
• Magnetic Dipole Moment
• A current loop experiences no net magnetic force in a uniform field
• But it does experience a torque
Summary

The magnetism of materials is due tothe magnetic dipole momentsof atoms, which arise from:

• the orbital motion of electrons
• and the intrinsic magnetic moment of each electron
Summary

Three classes of materials

• Diamagnetic M = –const • Bext, small effect (10-4)
• Paramagnetic M = +const • Bext small effect (10-2)
• Ferromagnetic M ≠ const • Bext large effect (1000)