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1
Chapters 27 and 25 (excluding 25.4)
2
Magnetism
Magnetism known to the ancients
Most Famous Magnet: Earth North=South! (today)
Seems to have flipped several times
Based on orientation of magnetic layers in the earth
Is Moving! From 1580 to 1820,
compass changed by 35o
||Bearth|| = 8 x 1022 J/T
S
N
3
Geomagnetism: It’s a life saver!
Sun and other galactic radiation sources emit charged particles
Magnetic fields divert charged particles Astronauts can get large radiation doses
Geomagnetic anomaly off of Tierra del Fuego
4
Origin of Geomagnetism
Uranium and other radioactive materials provide heat through alpha decay
This heat keeps the earth’s core (mostly iron) hot
The molten iron circulates
5
Broken Symmetry
There are no magnetic monopoles i.e the simplest magnetic system is a north pole-south pole system
Simplest Electric System
Simplest Magnetic System
6
A magnetic field does not diverge, its’ field line circulate
00 AdBB
qAdEE
allyMathematic
o
enclosed
o
enclosed
Gauss’s Law for Magnetic Fields
7
Magnetic Fields exerts a force on charged particles
Force is proportional to the charge,q, the velocity of the charge,v, and the strength of the magnetic field,BSince v, B, F are vectorsWe need a way to multiply a vector by a
vector and get a vector: cross-productF=qv x B||F||=qvB sin where is the angle
between v and B
8
Direction of Force
9
Units
Units of B = newtons/(coulomb* meter/second) Called Tesla (T) Coulomb/second called Ampere (A) T=N/(A*m) cgs units are gauss (G)
where 1 T = 104 G Earth’s magnetic field at any point is about 1 G Largest magnetic field is 45 T (explosion-
induce about 120 T)
10
Magnetic Flux
AdBB
Magnet flux through a closed surface=0 This is the field lines through a surface
Units=weber (Wb) and 1 Wb=1 T*m
11
Motion of Charged Particles in a Magnetic Field
Since F is perpendicular to v, there is no acceleration but it does change the direction
A particle moving initially perpendicular to B remains perpendicular to B
Particle’s path is a circle traced out with a constant speed, v
0
W
sovFthenBvqFIf
dt
rdvandrdFW
ort
xvandxFW
12
Mathematically
qB
mvR
qvBr
vm
qvBFr
vmF
2
2
m
qBf
m
qBf
fqB
mT
qB
m
v
rbut
v
rT
2
2
12
2
R is the radius of the charged particles path
is the angular frequency of the particlef is called the cyclotron frequency
13
Combined Force: Lorentz Force
If there is a static electric field, E, and a static magnetic field, B, a force is exerted on the particle equivalent to
BvqEqF
14
Velocity selector
Let E and B be perpendicular as shown below. We will solve for the velocity of particles are in
equilibrium (F=0).
B
Ev
qvBqE
qvBqEF
BvqEqF
0
15
Leaving Electrostatics
Electrostatics meant charges did not move
We will consider “steady” currents Steady currents are
constant currents Current: a stream of
moving charges
t
dtidqq
dt
dqi
0
16
Units
Ampere (A) = Coulomb/second (C/s)1 A in two parallel straight conductors
placed one meter apart produce a force of 2x10-7 N/m on each conductor
17
Can’t we all get along? (Blame Benjamin Franklin)
For physicists: The current arrow is drawn in the direction in which the
positive charge carriers would move Positive carriers move from positive to negative
For engineers: The current arrow is drawn in the direction in which the
negative charge carriers would move Negative carriers move from negative to positive
A negative of a negative is a positive so at the end of the day, we should all agree.
(Technically speaking, the engineers have it right.)
18
Current Density
Ad
AdJi
A
Ad
If the current is uniform and parallel to dA then i=JA or J=i/A
19
At the speed of what?
When a conductor has no current, the electrons drift randomly with no net velocity
When a conductor has a current, the electrons still drift randomly but they tend to drift with a velocity, vd in a direction opposite of the electric field
Drift speed is TINY (about 10-5 to 10-4 m/s) compared to the random velocity of 106 m/s
So if the electrons only move at 0.1mm/s then why do the lights come on so fast?
20
Charge carrier density
Let n=number of charge carriers/volume
If wire has cross-sectional area, A, and length, L, then volume = AL
Total number of charges, q=n(AL)e
Let t be the time that the charges traverse the wire with drift velocity, vd, this must be t=L/vd
d
d
d
vneJ
A
iJif
nAev
vL
eALn
t
qi
)(
Charge carrier current density
21
Resistivity and Ohm’s Law
Each material has a property called resistivity, , which is defined as =E/J where E is the electric field and J is the
charge density (actual definition of Ohm’s law) Units: (V/m)/(A/m2)=*m
The reciprocal of resistivity is conductivity, . J=E
Materials are “ohmic” when is constant If materials do not depend on this simple relation,
then the material is non-ohmic
22
Resistance
“resistance” to current flow How much voltage required to make
current flow
Units: ohm =V/A () Symbol
i
VR
23
Relationship between Resistance and Resistivity
A
LR
or
L
AR
L
A
i
V
AiL
V
J
E
thenA
iJ
L
VEthenLdand
d
VEif
24
Ohm’s Law
A current through a device is always proportional to the potential difference applied
V
i
resistor
V
i
diode
Both obey V=iR but the resistor obeys Ohm’s law while the diode does not
25
Power in resistors
R
VP
R
ViIf
RiPiRVIf
Vidt
dqVP
so
VdqdWdt
dWP
2
2
26
Band Theory of Solids
Electrons are restricted to certain energy levels: they are “quantized” “quantized” think “pixilated” Electrons can occupy any level but cannot have an energy between levels
Proximity of the atoms squeezes these levels into a few bands
Conduction Band
BandValence
Band Gap
Band represents many energy levels in close proximity
Conduction Band
BandValence
Conduction Band
BandValence
Conductor Insulator Semiconductor
27
Force Law from current perspective
q=i*t For a length of wire, L, with drift velocity vd,
then t=L/vd so q=i*L/vd F=qv x B or F=qvB sin In the case of the wire, v=vd so
F=(i*L/vd)*vdBsin F=iL x B
Where ||L|| is the length of the wire and the direction of L points in the direction of current flow
For each infinitesimal piece of wire dL, has a force, dF exerted on it by B : dF=I dL x B
28
Force and Torque on a Current Loop
While this seems an academic exercise, its importance cannot be overstated.
This is the basis of both:Electrical motorPower generation
Thus, its results impact us immenselyWe would die without it.
29
Diagram
B
30
Forces
F=iL x B For sides length a
Always perpendicular to B (out of page) F=iab
Because of this: a a
the forces have opposite directions on opposite sidesF+ F-
For sides length b Their angle w.r.t. to B changes as the loop moves F=ibBsin(900-)=ibBcos
Bb
31
Directions
For length a, the forces are in the x-direction (+x-hat and –x-hat)
For length b, the forces are in the y-direction
So the net FORCE is zero
But not the net TORQUE!
32
Torques
Recall =r x FFor length b sides, their line of common
action is through the center and thus, their net torque is zero.
33
Sides of length, a, have a net torque
As shown in the figure on the right, the vector torques for both sides of length a are in the +y-direction
The torque is rFsin Where ||r||=b/2 F=iaB
=2(iaB)(b/2)sin Area=a*b=A
=iABsin
Fr
34
Magnetic Moment,
The product of iA is called the magnetic moment and is a vector quantity =i A n
Where n is normal to the area of the current loop
Since = x B, this behavior is similar to that of an electric dipole ( = p x E) Thus, is sometimes called a magnetic dipole You might expect that the potential energy would
have the form of U=-B
35
Magnets on an atomic level
Think of an electron as a charge orbiting the nucleus
This is a charge moving through space at a constant angular velocity so essentially i=q*v where v=r .and r=electron orbital radius.
So this is a small current loop with area=*r2
Thus atoms can experience torques and forces when subjected to magnetic fields
36
Hall Effect Assume a current i is flowing in
the positive x direction along a copper strip (as shown on the right)
A static magnetic field is directed into the page
B forces the negative charge carriers to the right
Eventually, the right side is filled with negative charges and the left side is depleted which sets up a potential difference
An electric field is produced The electric field is
proportional to the magnetic field which produces it and the current
In the next chapter, we will learn how the Hall effect is used to measure currents.
i