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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