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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

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

An electric field is a disturbance in space caused

by electric charge. A magnetic field is a

disturbance in space caused by moving electric


An electric field creates a force on electric charges.

A magnetic field creates a force on moving electric


magnetic field and force
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 force5
Magnetic Field and Force

The force on a moving charge can

be written as

where B

represents the

magnetic field

magnetic field and force6
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 effect8


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



the hall effect9


The Hall Effect

Eventually, the induced electric force balances the

magnetic force:



t is the thickness

Hall coefficient

motion of charged particles in a magnetic field11
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 field12
Motion of Charged Particles in a Magnetic Field

Newton’s 2nd Law

So radius of circle is

motion of charged particles in a magnetic field13
Motion of Charged Particles in a Magnetic Field


the cyclotron period is

Its inverse is the cyclotron frequency

the biot savart law
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


As drawn, the field

is into the page

the biot savart law18
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


biot savart law example


Biot-Savart Law: Example

The magnetic field due to an infinitely long current

can be computed from the Biot-Savart law:


biot savart law example20
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


magnetic flux
Magnetic Flux

Just as we did for electric fields, we

can define a flux for a magnetic


But there is a profound difference

between the two kinds of flux…

gauss s law for magnetism
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 magnetism24
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…

ampere s law


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

ampere s law example




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.

ampere s law example27




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

placed about the current


ampere s law example28




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 current30
Magnetic Force on a Current

Force on each charge:

Force on wire segment:

n = number of charges

per unit volume

magnetic force on a current31
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
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:


magnetic moment
Magnetic Moment

A current loop experiences no net force

in a uniform magnetic field. But it does

experience a

F torque


The force is

F = IaB


magnetic moment35
Magnetic Moment

Magnitude of torque

where A= ab

For a loop with N turns, the

torque is

magnetic moment36
Magnetic Moment

It is useful to define a new vector

quantity called the magnetic dipole


then we can write the torque as

magnetic moment39
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


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
Types of Materials

Materials exhibit three types of magnetism:

  • paramagnetic
  • diamagnetic
  • ferromagnetic

Paramagnetic materials

  • have permanent magnetic moments
  • moments randomly oriented at normal temperatures
  • adds a small additional field to applied magnetic field
    • Small effect (changes B by only 0.01%)
  • Example materials
    • Oxygen, aluminum, tungsten, platinum

Diamagnetic materials

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

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


Example materials

  • high temperaturesuperconductors
  • copper
  • silver

Ferromagnetic materials

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

Ferromagnetic materials

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

have domains of randomly

aligned magnetization

(due to strong interaction

of magnetic moments of neighboring



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


In ferromagnets, some magnetization

will remain after the applied

field is reduced to zero,

yielding permanent magnets

Such materials exhibit


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

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

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)