Chapter 19

Magnetism

General Physics

Review – Magnetic FieldsReview – Magnetic FieldsELECTRIC FIELDSELECTRIC FIELDS From (+) to (–) From (+) to (–)

charges charges Field lines (electric flux)Field lines (electric flux) Start / End at chargesStart / End at charges NO loops! (cons. NO loops! (cons.

energy)energy) Force Law:Force Law:

F = q E (does work)F = q E (does work)

= d × E (elec. dipole)= d × E (elec. dipole)

MAGNETIC FIELDSMAGNETIC FIELDS From (N) to (S) polesFrom (N) to (S) poles

Field lines (magnetic flux)Field lines (magnetic flux) NO monopoles! (Start/End)NO monopoles! (Start/End) Loop (S) to (N) insideLoop (S) to (N) inside

Force Law: Force Law: ((× = sin × = sin )) F = q v × B (deflection)F = q v × B (deflection) F = B I LF = B I L (wire) (wire) = = × B (mag. dipole) × B (mag. dipole)

General Physics

Review – Right-hand ruleReview – Right-hand rule

Essence of a cross productEssence of a cross product F = q v × B F = q v × B

v B sin v B sin Force is perpendicular to Force is perpendicular to

both velocity and fieldboth velocity and field Need right-hand rule toNeed right-hand rule to

decide which directiondecide which direction Deflection doesn’t do workDeflection doesn’t do work

General Physics

Magnetic Fields IISections 6–10

General Physics

Motion of a Charged Particle in a Uniform Magnetic Field

Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field

The force is always directed toward the center of the circular path

The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

General Physics

Equating the magnetic and centripetal forces:

Solving for the radius r:

r is proportional to the momentum mv of the particle and inversely proportional to the magnetic field

Sometimes called the cyclotron equation

r

mvqvBF

2

qB

mvr

Motion of a Charged Particle in a Uniform Magnetic Field, cont

Active Figure: Motion of a Charged Particle in a Uniform Magnetic Field

General Physics

The Mass Spectrometer: Separating Isotopes

The cyclotron equation can be applied to the process of separating isotopes

Singly ionized isotopes are injected into a velocity selector

Only those isotopes with velocity v = E/B pass into the deflection chamber—Why?

Isotopes travel in different circular paths governed by the cyclotron equation—therefore different mass isotopes separate

Active Figure: The Mass Spectrometer

qB

mvr

General Physics

Magnetic SpectrometerMagnetic Spectrometerwith Drift (Ion) Chamberswith Drift (Ion) Chambers

2 sectors 2 sectors ×× 3 drift 3 drift chamberschambers

954 sense wires954 sense wires resolution 200 resolution 200 μmμm signal to noise 20:1signal to noise 20:1

8-coil toroid 8-coil toroid electromagnetelectromagnet

0.3 T maximum field0.3 T maximum field

General Physics

Particle Moving in an External Magnetic Field

If the particle’s velocity is not perpendicular to the magnetic field, the path followed by the particle is a spiral The spiral path is

called a helix

Active Figure: A Charged Particle with a Helical Path

General Physics

Charged Particles Trapped in the Earth’s Magnetic Field—Auroras

Charged particles from the Sun enter the Earth’s magnetic field

These particles move in spirals around the lines of magnetic field

This causes them to become trapped in the Earth’s magnetic field

An aurora is caused by these trapped charged particles colliding with atoms in the upper atmosphere—producing beautiful displays of light

General Physics

Charged Particles Trapped in the Earth’s Magnetic Field—Auroras

General Physics

Charged Particles Trapped in the Earth’s Magnetic Field—Auroras

General Physics

Charged Particles Trapped in the Earth’s Magnetic Field—Auroras

General Physics

Hans Christian Oersted

1777 – 1851 Best known for

observing that a compass needle deflects when placed near a wire carrying a current First evidence of a

connection between electric and magnetic phenomena

General Physics

Magnetic Fields – Long Straight Wire

A current-carrying wire produces a magnetic field

The compass needle deflects in directions tangent to the circle The compass needle points in

the direction of the magnetic field produced by the current

Active Figure: Magnetic Field Due to a Long Straight Wire

General Physics

Direction of the Field of a Long Straight Wire

Right Hand Rule #2 Grasp the wire in

your right hand Point your thumb in

the direction of the current

Your fingers will curl in the direction of the field

General Physics

Magnitude of the Field of a Long Straight Wire

The magnitude of the field at a distance r from a wire carrying a current of I is

µo = 4 x 10-7 T.m / A µo is called the permeability of

free space

2oIBr

General Physics

André-Marie Ampère

1775 – 1836 Credited with the

discovery of electromagnetism Relationship between

electric currents and magnetic fields

Mathematical genius evident by age 12

General Physics

Ampère’s Law

André-Marie Ampère found a procedure for deriving the relationship between the current in a wire and the magnetic field produced by the wire

Ampère’s Circuital Law B|| Δℓ = µo I Sum over the closed path

around the current I

Choose an arbitrary closed path around the current

Sum all the products of B|| Δℓ around the closed path

General Physics

Ampère’s Law to Find B for a Long Straight Wire

Sum over a closed circular path around current I

B|| Δℓ = µo I

Sum all products B|| Δℓ around the closed path

B·2r = µo I

The magnitude of the magnetic field a distance r from the wire

2oIBr

General Physics

Magnetic Field of a Current Loop

The strength of a magnetic field produced by a wire can be enhanced by forming the wire into a loop

All the segments, Δx, contribute to the field, increasing its strength

The magnitude of the magnetic field at the center of a circular loop with a radius R

2oIBR

General Physics

Magnetic Field of a Current Loop – Total Field

General Physics

Magnetic Field of a Solenoid

If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid

It is also known as an electromagnet since it acts like a magnet only when it carries a current

General Physics

Magnetic Field of a Solenoid, 2

The field lines inside the solenoid are nearly parallel, uniformly spaced, and close together This indicates that the field inside the

solenoid is nearly uniform and strong The exterior field is nonuniform,

much weaker, and in the opposite direction to the field inside the solenoid

General Physics

Magnetic Field in a Solenoid, 3

The field lines of the solenoid resemble those of a bar magnet – dipole magnetic field

General Physics

Magnetic Field in a Solenoid from Ampère’s Law

A cross-sectional view of a tightly wound solenoid

If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero

Apply Ampère’s Law to the blue dashed rectangle

The magnitude of the field inside a solenoid is constant at all points far from its ends

n is the number of turns per unit length

n = N / ℓ

nIB 0

General Physics

Magnetic Force Between Two Parallel Conductors

The force on wire 1 is due to the current in wire 1 and the magnetic field produced by wire 2

The force per unit length is: 1 2

2o I IF

d

General Physics

Force Between Two Conductors, cont

Parallel conductors carrying currents in the same direction attract each other

Parallel conductors carrying currents in the opposite directions repel each other

Active Figure: Force Between Long Parallel Wires

General Physics

Defining Ampere and Coulomb The force between parallel conductors can

be used to define the Ampere (A) If two long, parallel wires 1 m apart carry the

same current, and the magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current is defined to be 1 A

The SI unit of charge, the Coulomb (C), can be defined in terms of the Ampere If a conductor carries a steady current of 1 A,

then the quantity of charge that flows through any cross section in 1 second is 1 C

General Physics

Magnetic Effects of Electrons – Orbits

An individual atom should act like a magnet because of the motion of the electrons about the nucleus Each electron circles the atom once in about every 10-16

seconds This would produce a current of 1.6 mA and a magnetic

field of about 20 T at the center of the circular path

However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom

The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

General Physics

Magnetic Effects of Electrons – Spins

Electrons also have spin The classical model is

to consider the electrons to spin like tops

It is actually a quantum effect

General Physics

Magnetic Effects of Electrons – Spins, cont

The field due to the spinning is generally stronger than the field due to the orbital motion

Electrons usually pair up with their spins opposite each other, so their fields cancel each other That is why most materials are not

naturally magnetic

General Physics

Magnetic Effects of Electrons – Domains

In some materials, the spins do not naturally cancel Such materials are called ferromagnetic

Large groups of atoms in which the spins are aligned are called domains

When an external field is applied, the domains that are aligned with the field tend to grow at the expense of the others This causes the material to become

magnetized

General Physics

Domains, cont Random alignment (left) shows an

unmagnetized material When an external field is applied, the

domains aligned with B grow (right)

General Physics

Domains and Permanent Magnets

In hard magnetic materials, the domains remain aligned after the external field is removed The result is a permanent magnet

In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state

When a ferromagnetic core is placed inside a current-carrying loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field