26. Magnetism: Force & Field

2

Topics

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

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Introduction

An electric field is a disturbance in space causedby electric charge. A magnetic field is adisturbance 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.

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Magnetic Field and Force

It has been found that the magnetic force depends on the angle between the velocityof the electric charge and the magnetic field

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F v Bq

The force on a moving charge canbe written as

where B represents themagnetic field

Magnetic Field and Force

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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

Magnetic Field and Force

The Hall Effect

8

h

The Hall Effect

Consider a magnetic field into the page and a currentflowing from left to right.

Free positive charges will be deflected upwards and free negative chargesdownwards.

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The Hall Effect

dqv B qEHallVoltage

Eventually, the induced electric force balances themagnetic force:

h1

H

d

V Eh

IBv Bh

tnq

Hall coefficient t is the thickness

Motion of Charged Particles in a Magnetic Field

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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 ofthe point charge.

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Newton’s 2nd Law

2

F

qvB

am

rm

v

So radius of circle is

mv pr

qB qB

Motion of Charged Particles in a Magnetic Field

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Since,

2 2r mT

v qB

the cyclotron period is

mvr

qB

Its inverse is the cyclotron frequency

Motion of Charged Particles in a Magnetic Field

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The Van Allen Belts

15Wikimedia Commons

Origin of the Magnetic Field

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The Biot-Savart Law

A point charge produces an electric field.When the charge moves it produces amagnetic field, B:

02

ˆ

4

q rvB

r

0 is the magneticconstant:

70

7 2

4 10 T m/A

4 10 N/A

As drawn, the fieldis into the page

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The Biot-Savart Law

When the expression for B is extendedto a current element, IdL, we get the Biot-Savart law:

02

ˆ

4

I dLdB

r

r

02

ˆ

4

I dL rB

r

The total field is found by

integration:

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Biot-Savart Law: Example

dL

2 2r x y y

I r̂

P

The magnetic field due to an infinitely long current can be computed from the Biot-Savart law:

0 ˆ4

IB k

y

0 02 2

ˆ sin ˆ4 4

I IdL r dLB k

r r

x

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Biot-Savart Law: Example

Note: if your right-hand thumb points in thedirection of the current, your fingers will curl in thedirection of the resulting magnetic field

0

4

IB

y

I

Laws of Magnetism

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Magnetic Flux

Just as we did for electric fields, we can define a flux for a magnetic field:

d B dA

B

dA

But there is a profound differencebetween the two kinds of flux…

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Gauss’s Law for Magnetism

Isolated positive and negative electriccharges 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 themagnetic flux into the surface is exactlyequal to the flux out of the closed surface

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Gauss’s Law for Magnetism

This yields Gauss’s law for magnetism

Closed Surface

0B dA

Unfortunately, however, because this lawdoes not relate the magnetic field to itssource it is not useful for computingmagnetic fields. But there is a law that is…

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Ampere’s Law

Encircled

Closed Lo

0

op

B dr I

I

B

dr

If one sums the dot product arounda closed loop that encircles a steady current

I then Ampere’s law holds:

B dr

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

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Ampere’s Law: Example

What’s the magnetic field a distance z above aninfinite current sheet of current density per unitlength in the y direction? From symmetry, the magnetic

field must point in thepositive y directionabove the sheet and inthe negative y directionbelow the sheet.

x

yz

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Ampere’s Law: Example

Ampere’s law states that the line integral of themagnetic field along any closed loop is equal to 0

times the current it encircles:

x

yz

Encircled

Closed Lo

0

op

B dr I

Draw a rectangularloop of height2a in z and length bin y, symmetricallyplaced about the currentsheet.

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Ampere’s Law: Example

The only contribution to the integral is from the upperand lower segments of the loop. From symmetry themagnitude of the magnetic field is constant and the

same on both segments. Therefore, the integral is just 2Bb. The encircled current is I = b. So, Ampere’s

law gives 2Bb = 0 b and therefore B = 0 / 2

x

yz

Magnetic Force on a Current

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Force on each charge:dq v B

Force on wire segment:

d

d

F v B ALq

qn v A L B

n

LI B

Magnetic Force on a Current

n = number of charges per unit volume

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Magnetic Force on a Current

Note the direction of the force onthe wire

For a current element IdL the force is

IdLdF B

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Magnetic Force Between Conductors

0 1 2 22 2

I I dldF

d

dF Idl B

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

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Magnetic Moment

A current loop experiences no net forcein a uniform magnetic field. But it does

experience aF torque

F

B

The force isF = IaB

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Magnetic Moment

Magnitude of torque

sin sinF Bb bIa sinI BA

where A = ab

For a loop with N turns, the torque is sinI BN A

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Magnetic Moment

It is useful to define a new vectorquantity called the magnetic dipolemoment

n̂ANI

then we can write the torque as

ˆ B

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2ˆ ˆnAI R nI

Example: Tilting a Loop

ˆ B

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2ˆ ˆnAI R nI

Example: Tilting a Loop

ˆ B

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Magnetic Moment

The magnetic torque that causes the dipole to rotate does work and tends todecrease the potential energy of the magnetic dipole

If we agree to set the potential energy to zeroat 90o then the potential energy is given by

U B

B

Magnetization

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Magnetization

Atoms have magnetic dipole moments due to

orbital motion of the electrons

magnetic moment of the electron

When the magneticmoments align wesay that the materialis magnetized.

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Types of Materials

Materials exhibit three types of magnetism:paramagnetic

diamagnetic

ferromagnetic

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Paramagnetism

Paramagnetic materials have permanent magnetic moments

moments randomly oriented at normal temperatures

adds a small additional field to applied magnetic field

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Paramagnetism

Small effect (changes B by only 0.01%)

Example materials Oxygen, aluminum, tungsten, platinum

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Diamagnetism

Diamagnetic materialsno permanent magnetic moments

magnetic moments induced by applied magnetic field B

applied field creates magnetic moments opposed to the field

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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

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Diamagnetism

Example materials high temperature superconductors

coppersilver

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Ferromagnetism

Ferromagnetic materialshave permanent magnetic moments

align at normal temperatures when an external field is applied and strongly enhances applied magnetic field

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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)

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Ferromagnetism

Applying a magnetic field causes domainsaligned with the applied field to grow at the expense of others that shrink

Saturation magnetization is reachedwhen the aligned domains have replaced all others

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Ferromagnetism

In ferromagnets, some magnetization will remain after the applied field is reduced to zero, yielding permanent magnets

Such materials exhibithysteresis

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Summary

Magnetic ForcePerpendicular to velocity and fieldDoes no work Changes direction of motion of charged

particle Motion of Point Charge

Helical path about field

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Summary

Magnetic Dipole MomentA current loop experiences no net magnetic

force in a uniform field

But it does experience a torque

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Summary

The magnetism of materials is due to the magnetic dipole moments of atoms, which arise from:the orbital motion of electrons

and the intrinsic magnetic moment of each electron

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Summary

Three classes of materialsDiamagnetic M = –const • Bext,

small effect (10-4)Paramagnetic M = +const • Bext

small effect (10-2)

Ferromagnetic M ≠ const • Bext

large effect (1000)