Transcript Chapter 27-27.5

Chapters 27 and 25
(excluding 25.4)
1
Magnetism


Magnetism known to the
ancients
Most Famous Magnet:
Earth

North=South! (today)

Seems to have flipped
several times


Based on orientation of N
magnetic layers in the
earth
Is Moving!


S
From 1580 to 1820,
compass changed by
35o
||Bearth|| = 8 x 1022 J/T
2
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
3
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
4
Broken Symmetry
There are no magnetic monopoles i.e the
simplest magnetic system is a north polesouth pole system
Simplest
Electric
System
Simplest
Magnetic
System
5
A magnetic field does not diverge,
its’ field line circulate
Mathematically
  enclosed
  qenclosed
E 
  E  dA 
o

 
  B  0   B  dA  0
o
Gauss’s Law for Magnetic
Fields
6
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,B
Since v, B, F are vectors
 We need a way to multiply a vector by a
vector and get a vector: cross-product

F=qv x B
 ||F||=qvB sin f where f is the angle
between v and B

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Direction of Force
8
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 (explosioninduce about 120 T)

9
Magnetic Flux
 
 B   B  dA

Magnet flux through a closed surface=0
 This is the field lines through a surface

Units=weber (Wb) and 1 Wb=1 T*m
10
Motion of Charged Particles in a
Magnetic Field
 
W  F  x and

 x
v
t
or
 

W  F  dr and v 

 
If F  qv  B then
W 0

dr
dt
 
F  v so

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
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Mathematically
v2
F m
r
F  qvB
v2
m  qvB
r
mv
R
qB
R is the radius of the
charged particles
path
2r
T
v
r m
but

v qB
2m 1
qB
T

 f 
qB
f
2m
qB
  2f 
m
 is the angular frequency of the particle
f is called the cyclotron frequency
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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


 
F  qE  qv  B
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Velocity selector


Let E and B be perpendicular as shown below.
We will solve for the velocity of particles are in
equilibrium (F=0).


 
F  qE  qv  B
F  qE  qvB
0   qE  qvB
E
v
B
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Leaving Electrostatics

Electrostatics meant
charges did not
move
 We will consider
“steady” currents


Steady currents are
constant currents
Current: a stream of
moving charges
dq
i
dt
t
q   dq   i dt
0
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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
16
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.)
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Current Density
 
i   J  dA
q
q

dA
q

dA
q
A
If the current is uniform
and parallel to dA then
i=JA or J=i/A
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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?
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Charge carrier density




Let n=number of charge
carriers/volume
If wire has crosssectional 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
q n( AL)e
i 
 nAevd
L
t
vd
i
if J 
A


J  nevd
Charge carrier current
density
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Resistivity and Ohm’s Law

Each material has a property called resistivity,
, which is defined as



The reciprocal of resistivity is conductivity, s.


=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)=W*m
J=sE
Materials are “ohmic” when  is constant

If materials do not depend on this simple relation,
then the material is non-ohmic
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Resistance

“resistance” to current flow

How much voltage required to make
current flow
V
R
i

Units: ohm =V/A (W)
 Symbol
22
Relationship between Resistance
and Resistivity
if E 
V
d
and d  L then E 
V
L
i
J
A
then
V
E L V A
A
  
R
i
J
i L
L
A
or
L
R
A
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Ohm’s Law

A current through a device is always
proportional to the potential difference
applied
i
i
V
V
resistor
Both obey V=iR but the
resistor obeys Ohm’s law
while the diode does not
diode
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Power in resistors
dW
P
dt
dW  Vdq
so
dq
P V
 Vi
dt
If V  iR  P  i 2 R
2
V
V
If i   P 
R
R
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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
Band
represents
many energy
levels in
close
proximity
Conduction
Band
Conduction
Band
Valence
Band
Valence
Band
Conduction
Band
Valence
Band
Band Gap
Conductor
Insulator
Semiconductor
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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 q
 In the case of the wire, v=vd so


F=(i*L/vd)*vdBsin q
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
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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 motor
 Power generation


Thus, its results impact us immensely

We would die without it.
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Diagram
B
29
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 sides
F+
F-
For sides length b


Their angle w.r.t. to B changes as the loop moves
F=ibBsin(900-f)=ibBcosf
B
b
f
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Directions

For length a, the
forces are in the xdirection (+x-hat and
–x-hat)
 For length b, the
forces are in the ydirection
 So the net FORCE is
zero
But not the net
TORQUE!
31
Torques
Recall t=r x F
 For length b sides, their line of common
action is through the center and thus,
their net torque is zero.

32
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 +ydirection
The torque is rFsinf



t=2(iaB)(b/2)sinf


Where ||r||=b/2
F=iaB
Area=a*b=A
f
r
f
F
t=iABsinf
33
Magnetic Moment, m

The product of iA is called the magnetic
moment and is a vector quantity

m =i A n


Where n is normal to the area of the current loop
Since t = m x B, this behavior is similar to that
of an electric dipole (t = p x E)


Thus, m is sometimes called a magnetic dipole
You might expect that the potential energy would
have the form of U=-mB
34
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
35
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
i
In the next chapter, we will learn
how the Hall effect is used to
measure currents.
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