Electricity

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Electricity Magnetism

• Seb Oliver
• Lecture 17
• Solenoids
• Gausss Law in Magnetism
• Micro-magnetism

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Summary Lecture 16

• Amperes Law
• Easier to use than Biot-Savart Law in many cases
• Examples
• Long-wire
• Inside wire
• Toroid
• Solenoid

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Today

• Solenoid
• Gausss Law in Magnetism
• Microscopic description of magentism

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Solenoid
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Infinitely Long Solenoid
Wire carrying a current of I0 wrapped around with
n coils per unit length
Zoom looks very similar to the toroid with a very
large radius
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Toroidal Coil Revisited
I0
Toroid has N loops of wire, carrying a current I0
Central radius R circumference is 2pR
Number of coils per unit length n is
From earlier
Independent of R
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Infinitely Long Solenoid
Wire carrying a current of I0 wrapped around with
n coils per unit length
Field at centre is same as torus of infinite
radius
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Magnetic Field Lines

• Field lines between adjacent wires cancel
• Field lines down axis of solenoid reinforce each
  other
• Field lines outside solenoid cancel

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Magnetic field from a solenoid
Similar to a bar magnet
Weak Field off-axis
Strong Field along axis
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Gausss Law in Magnetism
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Gausss Law in Magnetism

• Magnetic Flux
• Gausss Law
• Magnetic Monopoles

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Electric Flux

• Magnetic flux is defined similarly to electric
  flux

Electric flux
If E to A
If E not to A
In vector notation
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Magnetic Flux
Magnetic flux
If B to A
If B not to A
In vector notation
Flux proportional to number of Field lines
Magnetic Flux
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Gausss Law in Electrostatics
Charge contained within the surface
Total Flux through a surface
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Gausss Law in Magnetism
Total Flux through a surface
Gausss Law

• All fields lines must pass in and out of a
  surface
• No Magnetic monopoles

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Microscopic Magnetism
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Generation of Magnetic Fields

• Magnetic Field produced by current loop
• Torque on current loop in magnetic field

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Magnetic Field from Current Loop
at centre
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Force on Current Carrying Loop
F 0
L
L
F -ILB
L
F ILB
B
L
F 0
Total Force is zero
However Torque is not zero
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Torque on Current Carrying Loop
However Torque is not zero
B
L
L
L
Since LB perpendicular L
L
Where we have defined a magnetic moment m IA
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Two Current Loops
Loop 2 feels a torque which is acting to turn the
loop so that the magnetic moments align
Loop 1 has a magnetic moment which produces a
magnetic field
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Many loops Magnetic Moments
Before
After
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Many loop Magnetic Field
Before
After
Little or no nett magnetic field
Stronger Magnetic Field
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Microscopic Description of Magnetism Magnetic
moment of electron
Bohr model of the atom
Electron whizzing around nucleus
path
current
speed
Magnetic moment
Period
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Angular Momentum
Electrons are usually described in terms of
angular momentum
Angular momentum (L mvr) is the angular
equivalent of linear momentum (p mv)
Angular momentum is conserved (unless a torque is
acting) thus a ballerina goes faster if she pulls
her arms in
Magnetic moment is proportional to spin
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Microscopic Description of Magnetism

• Magnetic moment of an atom is related to spin of
  the electrons
• A net magnetic moment can be produced in
  materials where the net spin is non-zero
• Magnetic moments of lots of atoms will tend to
  line up
• Producing a Macroscopic Magnetic Field

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Magnetic Materials
Magnetic Moment per unit volume

• Paramagnetic
• M ? applied magnetic field B
• Ferromagnetic
• m align even in a weak magnetic field
• Strong magnetic materials
• Diamagnetic
• No permanent m
• M ? -B

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Diamagnetic Materials
2 electrons spinning in opposite directions L-,
L
Applying a magnetic field produces a force on the
electrons F v ? B
B
This cannot change the speed, but can change the
radius of the orbit
This in turn affects the angular momentum
r decreases
r increases
L decreases
L increases
L increases
m decreases
m decreases
m increases
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Summary Lecture 17

• Solenoid
• Field lines around a solenoid
• strong inside, weak outside
• Gausss Law in Magnetism (not covered Aut. 2002)
• All flux lines entering a surface leave it
• No magnetic monopoles
• Microscopic description of magnetism
• Electrons orbiting nuclei in atoms act like
  charge loops producing small magnetic fields
• These fields can align either spontaneously
  (Ferromagnetism) or in the presence of an
  external magnetic field (Paramagnetism)