1

Chapters 27 and 25 (excluding 25.4)

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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

qq

qq

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

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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 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

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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 motorPower generation

Thus, its results impact us immenselyWe would die without it.

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Diagram

B

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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

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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.

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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

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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