26. magnetism: force & field. 2 topics the magnetic field and force the hall effect motion of...
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26. Magnetism: Force & Field
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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
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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)