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BELLRINGER
Compare and explain in
complete sentences and formulas
what is the unit for nuclear force.
Homework due
tomorrow
WHAT IS THE LAW OF
CONSERVATION OF ENERGY?
GIVE EXAMPLES.
There are Four Fundamental Forces:
These are
responsible for
all we see
accelerate
1) The Electromagnetic Force
(We’ll study it this term)
These act
over a very
small range
3) The Strong Nuclear Force
2) The Gravitational Force
4) The Weak Nuclear Force
The Unification of Forces
Physicists would love to be able to show someday
that the four fundamental forces are actually
the result of one single force that was present
when our universe began.
Superstring Theory is an interesting and
promising possibility in this quest:
Web Links: Superstring Theory
The Elegant Universe
The Fabric of the Cosmos
Recent Physics Discovery!
Now let’s review the gravitational force…
Any two masses are attracted by equal and opposite
gravitational forces:
m1
F
-F
m2
r
where……
m1m2
FG 2
r
Newton’s Universal
Law of Gravitation
G=Universal Gravitation Constant =
6.67x10-11 Nm2/kg2


This is an Inverse-Square force
Gravity is a very weak force
If an atom has the
same amount of
+ and - charge
Neutral
(no net charge)
If it’s missing electrons
net + charge
If it has extra electrons
net - charge
atom
-
-
++++
glass
silk
-
(rub)
-
---plastic
fur
Web Links: Static Duster
New Carpet
(rub)
Ex:
If you rub a balloon against your hair,
which ends up with more electrons, the
balloon or your hair?
Opposites
Charges
Attract
Like
Charges
Repel
Conductors
(such as metals, tap or salt
water, and the human body)
are good at conducting away
any extra charge.
Metal:
“free electrons”
Touching it
with your
hand will
discharge it
Insulators
(like plastic, rubber,
pure water, and glass)
will not conduct away
extra charge.
Use rubber gloves
in the lab
Grounding
-Object is
-
+ +
-
+
discharged or
“grounded”
+ -+
-
-
-+ -
+
+
-
+
The earth is a huge reservoir of
positive and negative charge
Induced Charge
(Charging by Induction)
+
+
+
+
What happens when you
bring a neutral metal
object near a positively
charged object?
What happens when you
bring a neutral metal
object near a negatively
charged object?
-
Web Links: Charging by Induction 1
Charging by Induction 2
Electric Current
wire
-
-
- electrons
Current
Electric current is in the
direction that positive charge
carriers “would” move
why?
ask Ben Franklin
Current = Charge per Time
q
I
t
I  q t
Amperes (A) Coulombs (C) seconds (s)
SI units
Remember, opposite charges attract:
q1 and q2
may
represent
lots of
extra or
missing
electrons
and like charges repel:
How much force do q1 and q2 exert on each other?
F  electrostatic force  k
Web Link:
Orbiting electron
q1 q2
r2
Coulomb’s
Law
k = electrostatic constant
= 8.99 x 109 Nm2/C2
Notes on Coulomb’s Law
1) It has the same form as the Law of Gravitation:
Inverse-Square Force
2) But…
(can you spot the most basic difference
between these two laws?)
3) The electrostatic constant (k) in this law is derived
from a more fundamental constant:
k
1
4 0
0= permittivity of free space
= 8.85 x 10-12 C2/Nm2
4) Coulomb’s Law obeys the principle of superposition
Web Links: Coulomb force, Releasing a test charge
Ex:
+q
-q
r
+q
r
What is the direction of the net force on the
charge in the middle ?
What about the charge on the left?
What about the charge on the right?
Ex:
q2
q1= +4.0 C
.15 m
q2= -6.0 C
q3= -5.0 C
73°
q1
q3
.10 m
Find the net force on charge q1
Smallest possible amount of charge:
1 extra electron: q = -1.60 x 10-19 C
1 missing electron: q = +1.60 x 10-19 C
For any charge q:
=e
= elementary
charge
…
q = ne , where n = 1, 2, 3, etc…
Charge is quantized
Also:
Charge is conserved
Ex:
1.0 cm
electron
+
proton
Calculate both the gravitational force and the
electrostatic force, and compare their magnitudes.
Electric Fields
Field – A set of values that defines a given property
at every point in space
Temperature Field:
Elevation Field:
Both of these examples are scalar fields
We need to look at a vector field
Wind
Notice that the wind vectors each have
magnitude and direction
This is an example of a vector field
Here is an animated example: Wind Map
Electric Field (E) – A vector field surrounding
a fixed, charged object that indicates the force
on a positive test charge (q0) placed nearby
+ +
+ + +
+ +
+ +
fixed, charged object
+
test charge
+
Draw the Electric Field vector at the position of
the test charge.
 Draw the Electric Field vectors at several other
positions surrounding the fixed, charged object.

Web Link: Force Fields
F
E
q0
The Electric Field is defined as the
Force per unit Charge at that point
Notes on E-field
1) The E-field points in the direction of force on a
positive test charge
2) If a negative charge were placed in the E-field,
what do you suppose would happen?
3) The E-field is a property of the fixed charges only
(it is independent of the test charge)
4) E-fields add as vectors
5) Given the E-field value at a certain point, we can
calculate the force F on any charge q0 placed there:
F
E
q0
F = q0E
Ex:
.10 m
+
+
q0 = 1.0 C
(test charge)
q = 2.0 C
(fixed charge)
a) Find the force on the test charge using
Coulomb’s Law
b) Find the electric field at the position of
the test charge
c) Could you have answered part b without
knowing the value of the test charge?
E=?
r
q
kq
E 2
r
Electric Field at a distance r
from a point charge q
Electric Field Lines
-represent symmetric paths of a positive test charge
+
The number of lines is arbitrary, as long as
they are symmetric

The density of lines represents the strength
of the Electric field

What would the Electric field lines look like if
there was a negative charge at the center?

What do you think the Electric Field lines would
look like for…
A large (), charged,
non-conducting sheet?
A charged, non-conducting
sheet that is not infinite?
+ +
+ + +
+
+
+ + +
+ +
+ ++
+
++ +
+
+
-
+ +
- + +
+
+
- ++ +
-
Two oppositely
charged plates?
(called a parallel
plate capacitor)
+
The Electric Field Lines for 2 Equal Charges:
The Electric Field Lines for 2 Opposite Charges
(called an Electric Dipole):
Web Links: Electric Field Lines, Releasing a test charge
Charged Conductors
Any excess charge ends up on the surface
of a conductor, independent of its shape
Why do you think this happens?
What happens to a neutral conductor placed
in an external electric field?
E=0
At equilibrium, the Electric
Field at any point within a
conducting material is zero.
“Shielding”
Faraday Cage: an example of shielding
Consider two charged spheres, one having three
times the charge of the other. Which force diagram
correctly shows the magnitude and direction of the
electrostatic forces?
a)
b)
c)
+
+
++
+
++
+
+
+
++
+
++
+
+
+
++
++
++
d)
+
+
++
+
++
+
+
+
++
+
++
+
+
+
++
++
++
e)
f)
Recall…
Gravitational
Potential
Energy
or
Elastic Potential Energy
Now…
+
+ + +
+ +
+ +
+
+ +
-
Electric
Potential
Energy
(EPE)
Only Conservative Forces have an associated PE
Recall:
PEgrav = mg(h) = -(Work done by gravity)
Similarly:
-
EPE = -(Work done by electrostatic force)
= - (Fcos)s
Force
displacement
-
angle between F and s
EPE = -W = -(Fcos)s
+ +
++ +
+ + ++
++
Ex:
2.0 m
proton
+
+
Uniform Electric Field
E = 4.0 N/C
a) Find the force on the proton.
b) Find the work done by that force as the proton
moves 2.0 m.
c) Find the change in EPE as it moves 2.0 m.
d) Find the change in EPE if an electron were to
move through the same displacement.
Work is Path Independent for conservative forces:
Ex: Gravity
Ex: Electric Field
path 1
path 2
path 1
path 2
Work done by gravity
on path 1
=
Work done by gravity
on path 2
Work done by
electrostatic force
on path 1
=
Work done by
electrostatic force
on path 2
EPE is a type of mechanical energy, like…
++
+
Kinetic Energy (KE) = ½ mv2
Rotational Kinetic Energy (KER) = ½ I2
Gravitational Potential Energy (PEgrav) = mgh
Elastic Potential Energy(PEelast) = ½ kx2
= Total Mechanical Energy (E)
is conserved if there are no non-conservative
forces present (ie friction).
+
Ex:
1.0 m
proton
+
+
Uniform Electric Field
E = 150 N/C
A proton released from rest into this electric field will
be going how fast after traveling a distance of 1.0 m ?
Can you think of two different methods to use
in solving this problem?
Do they yield the same answers?
In both previous examples,
we saw that…
EPE  q
E
q
2q
Twice the charge has
twice the EPE
We would like to have a new quantity that
describes the “Potential” at various points in the
electric field independent of the charges in it:
EPE
= EPE per charge
Electric Potential  V 
q0
SI Unit = J/C = 1 Volt
Also called Potential or Voltage
From the definition of Electric Potential, we
can show that when a charge is moved from
one point to another in an electric field:
1
E
2
Work done
Charge
= - that was
by the
Electric Field
moved
Difference in Potential
between its old and
new positions
W = -q0(V)
Let’s make sure that we understand the difference
between Potential and Electric Potential Energy:
E
V (in Volts) = Potential
a property of a certain position in
an Electric Field with or without
charges placed there
EPE (in Joules) = Electric Potential Energy
a property of charges placed at a certain
position in an external Electric Field
Web Link: EPE vs Potential
+
-
E
We now have a new SI
unit for Electric Field:
Volts / meter
Ex
E = 3 N/C = 3 V/m
There is a force of 3 Newtons on
each 1 Coulomb of charge in the field
The Potential changes by 3 Volts for
every 1 meter of distance
We also have a new energy unit (not SI):
The electron-Volt (eV)
amount of energy gained (or lost) when 1 electron
moves through a potential difference of 1 volt
1V
Equipotential Surfaces
adjacent points at the same electric potential
E-field
Equipotential
Surface
Equipotential Surface
E-Field
Web Link: Equipotential surfaces
Equipotential Surfaces are 3-dimensional:
Notes on Equipotential Surfaces
1) Equipotential surfaces are always perpendicular
to Electric Field lines
Web Link: Electric Field Lines
2) If a charge moves on an equipotential surface,
the work done by the Electric Field is zero:
+
F
E-Field
s
Equipotential
Surface
Web Link: Equipotential surfaces
In the case of a Uniform Electric Field, it is
especially easy to calculate the potential difference
between equipotential surfaces:
+
+
+
+
-
E
Potential gets higher
in this direction
Potential gets lower
in this direction
E is in Volts/meter
E = V/s
V = E(s)
Ex:
E = 5.0 V/m
.30 m
Find the potential difference between
the plates.
In the lab, we could use a Voltmeter to
simply measure the potential difference:
This means there is a potential difference (V)
of 12 Volts between the terminals of the battery
Calculating the Potential due to a Point Charge
r
q
What is the Potential
at this point?
q
Vk
r
k = electrostatic constant
= 8.99 x 109 Nm2/C2
Notes:
1) Include the sign of q in your calculation! (+ or -)
2) Potential Difference can also be calculated:
q
q
k
V = V2 – V1  k
r2
r1
3) The equation can also be used for a charged sphere:
++
+
q
Total charge
+
+ r
Vk
++ ++
Distance from center
r
+
Van de Graff generator
Ex:
electron
a) Starting at 1.0 nm from the electron and moving
out to 5.0 nm from the electron, what is the
change in potential ?
b) What is the electric potential energy (in eV) of a
proton that is placed at a distance of 5.0 nm from
this electron?
c) What is the electric potential energy (in eV) of
another electron at a distance of 5.0 nm from this
one?
Calculating the Potential due to
Multiple Point Charges
+
+
What is the value of the Electric field
directly between equal charges?
What about the value of the
Electric Potential there?
Electric Potential is a scalar not a vector
V = V 1 + V 2 + V3 + …
(an algebraic sum, not a vector sum)
Ex:
+q
d
P
d
-q
d
-q
d
+q
Find the potential V at point P due to the
four charges.
Web Link: Complex Electric Field
Capacitor
a device that stores energy by
maintaining a separation between
positive and negative charge
(Symbol:
)
Circuit Board
Capacitor
Resistors
Parallel Plate Capacitor
+q
V
-q
-
-
This is called
“charging a
capacitor”
q = charge of the capacitor
V = potential difference of the capacitor
q and V are proportional:
q=CV
C = Capacitance
(a fixed property of each capacitor)
SI unit = 1 Farad (F) = 1 Coulomb / Volt
Dielectrics
electrically insulating materials
Capacitor without
a dielectric
Capacitor with
a dielectric
What happens to
the Electric Field?
The Electric Field magnitude is less in a dielectric
How much less depends on the
dielectric constant () of the material
Calculating the Capacitance (C) of a
parallel plate capacitor

A
A = plate area
d = plate separation
 = dielectric constant
C
d
Notice:
 0 A
d
(0= 8.85 x 10-12 C2/Nm2)

Capacitance is independent of both charge and voltage

Adding a dielectric increases the Capacitance
Web Links: Capacitance Factors, Lightning
How much Energy is stored by a capacitor?
Energy = ½CV2
Capacitance
Voltage
What’s the energy density in an Electric Field?
Energy 1
Energy Density 
  0 E2
Volume 2
* For any electric field
+q
-q
d
-q
+q
D
Consider a parallel plate capacitor with charge q and
plate separation d. Suppose the plates are pulled
apart until they are separated by a greater distance D.
The energy stored by the capacitor is now
1. greater than before
2. the same as before
3. less than before
Here’s a Web Link about
a huge capacitor and
what can be done with
all that stored energy:
Pulse Discharge Machine
Web Link: DC Electricity
V
Imagine
a wire:
-
-
-
+
E
Now imagine bending the
same wire into a loop:
Battery or other
emf source
+
V
-
-
emf – electromotive “force” – the
potential difference between the
terminals of an electric power source
-
Ex:
emf = 9 V
The current arrow points with
the “positive charge carriers”
I
+
+
+
Web Link: Conventional Current
q
current  I 
t
SI unit = Ampere(A) = 1 C/s
Notes on Current:
1) Remember: charge is conserved
2) Current is a scalar, not a vector
3) There are two types of current:
DC (direct current)
charge moves the same
direction at all times
AC (alternating current)
charge motion alternates
back and forth
Web Link: AC vs. DC
Ex:
A DC current of 5.0 A flows through this wire:
I
How much charge flows past this
point in 4.0 minutes?
Will the bird on the high voltage wire
be shocked?
V
Resistance  R 
I
applied voltage
resulting current
SI unit: Ohm () = 1 V/A
Web Link: Resistance
Resistor – a circuit component designed to provide a
specific amount of resistance to current flow.
(Resistor symbol:
)
Ex:
9V
1000 
Draw the circuit diagram, and calculate
the current in this circuit.
Resistance = R = a property of a given resistor
(Ex: 20  , 400  , etc.)
Resistivity =  = a property of a material used in
making resistors
Building Resistors
L

A
L
R 
A
(: SI unit = ·m)
Ex: Aluminum Power Lines
Consider an aluminum power line with a cross
sectional area of 4.9 x 10-4 m2 . Find the
resistance of 10.0 km of this wire.
Ex: Incandescent Light Bulb
Tungsten wire
radius .045 mm
120 V
I = 12.4 A
What is the length of the tungsten wire
inside the light bulb?
Web Link: Light bulb
V=IR
(IV)
“Ohm’s Law”
It works for
resistors:
Is it really a law ?
I
(IV)
V
What about other devices?
Light Bulb
Diode
I
“Ohm’s Law”
is not really
a Law!
I
V
(IV)
V
Power = P = IV
SI Unit = 1 Watt (W) = 1 J/s
Rate of energy transfer
If the device is a resistor:
V=IR
P = I V = I2 R
P = I V = V2/R
I=V/R
Energy dissipated
by the resistor as
thermal energy
Ex: Space Heater
120 V
1500 W
Heater
Find:
a) The resistance of the heater
b) The current through the heater
c) The amount of heat produced in 1 hour
…back to the difference between AC and DC:
Web Link: AC vs. DC
DC (
):
Ex:
Voltage
time
AC (
):
Ex:
time
Voltage
V = V0 sin ( 2  f t )
Voltage amplitude
frequency
radians
time
So what does AC current look like?
Typical
household
outlet:
Light bulb:
Resistance R
V0 = 170 V
f = 60 Hz
V V0 sin 2  f t 
I

R
R
= I0 = current
amplitude
I = I0 sin ( 2  f t )
I
t
Ex: Alarm Clock
V0 = 170 V
f = 60 Hz
How many times a day does the
current change direction?
AC Power
P=IV=?
peak values
Irms 
I0
2
Vrms 
V0
2
These are the
values that matter
Ex:
What is the
rms voltage?
V0 = 170 V
P = Irms Vrms
P = (Irms)2 R
P = (Vrms)2 / R
look familiar??
Ex: Speaker
If the power rating of the speaker is
55 Watts, and its resistance is 4.0 ,
what is the peak voltage?
Heating element of
resistance R
AC
generator
Resistors in Series
R1
R2
RS = R 1 + R 2
(RS > R1 , R2)
Resistors in Parallel
R1
1
1
1


R P R1 R 2
R2
(RP < R1 , R2)
R
R
Consider two identical resistors wired in series.
If there is an electric current through the
combination, the current in the second resistor is
1. equal to the current through the first resistor.
2. half of the current through the first resistor.
3. smaller than, but not necessarily half of the
current through the first resistor.
A
B
As more resistors are added to the parallel
circuit shown here, the total resistance
between points A and B
1. increases
2. remains the same
3. decreases
Ex:
For some holiday lights, if one bulb is bad, the
whole string goes out. For others, one bulb
can go out and the rest stay lighted. What is
the difference ?
I
Basic Circuit:
V
Series Circuit:
I
V

V
RS
I = V/R
Current (I) has the same value
everywhere in the circuit
current is
like a parade
R1
R2
I
R

VR1 + VR2 = VBattery
voltage is
like money

RS = R1 + R2

I = V/RS
Parallel Circuit:

I1
I2
V
R1
I3

Web Link:
Parallel Current
R2
?
V
I1 = I2 + I3

VBatt = VR1 = VR2
I1
RP
I1 = V/RP


I2 = V/R1

1
1
1


RP R1 R2
I3 = V/R2
Ex:
4
16 V
4
What is the series resistance?
Calculate the current in this circuit.
16 V
4
4
What is the parallel resistance?
Calculate the current in all branches
of this circuit.
Ex:
47 
V
28 
The current through the 47  resistor is .12 A
Calculate the voltage V of the battery.
Ex:
V
47 
28 
The current through the 47  resistor is .12 A
Calculate the current through the 28  resistor.
V
R1
In a series circuit,
the current is the
same through
each resistor
R2
In a parallel circuit, the
voltage is the same
across each resistor
V
R1
R2
Notice that the terminology will help us remember
how to measure current and voltage
Measure the voltage across a resistor:
Measure the current through a resistor:
You must break
the circuit to
measure current!
How to calculate the equivalent resistance
for a group of resistors:
Ex:
Find the
equivalent
resistance of
this circuit:
Kirchoff’s Rules
I) The Junction Rule
The sum of the currents entering any
junction is equal to the sum of the
currents leaving that junction.
Ex:
I2
I3
I1
I1+ I2+ I3= I4
I4
Web Link: Kirchoff’s 1st Law
II) The Loop Rule
The potential differences around
any closed loop sum to zero.
Web Link: Kirchoff’s 2nd Law
Ex:
+
R1
+
V
I2
-
-
I1
+
R2
-
This loop (clockwise):
+V - I2R1 - I2R2 = 0
I3
V = IR
+
R3
-
VR1 = I2R1
VR2 = I2R2
VR3 = ?
Write out the equations for
this loop and the outer loop
Here are the steps for applying Kirchoff’s Rules to
solve for unknown currents and voltages in a circuit:
Step 1) Label all the different currents in the circuit
I1, I2, I3, etc. (current direction is arbitrary)
Step 2) Apply the junction rule at each junction
(one junction will yield redundant information)
Step 3) Indicate which end of each device is + and -
-
+
I
+
-
Step 4) Apply the loop rule to each independent loop
Step 5) Solve the equations for the unknown quantities
Ex:
3.0 
8.0 V
4.0 
V
1.7 A
5.0 
Use Kirchoff’s rules to find
a) the remaining two currents in the circuit, and
b) the unknown voltage
Web Link: Building circuits
Capacitors in Circuits
Recall:

A
C
d
 0 A
CA
d
C  1/d
Capacitors in Series:
Capacitors in Parallel:
V
C1
CP = C1 + C2
C2
C1
V
C2
1
1
1


C S C1 C 2
Ex:
5V
8.0 F
4.0 F
6.0 F
a) Find the total capacitance of the circuit
b) Find the total charge stored on the capacitors
RC Circuits
Charging a Capacitor:
Web Link: RC Circuit I
At
t = 0: close the switch
First instant: I = V0/R
Then: I decreases as the
capacitor fills with charge
Charge on
capacitor
Finally:
I = 0, and Vcap = Vbattery = V0
full capacitor charge
q0 = CV0
Web Link:
RC Circuit II


q  q0 1  e

t

RC


RC = time constant = 
time
Discharging a Capacitor:
Web Link: RC Circuit I
The
capacitor starts out
fully charged to voltage V0
At
t = 0: close the switch
First
instant: I = V0/R
Then:
I decreases as the capacitor loses its charge
I = 0, and Vcap = 0
Charge on
capacitor
Finally:
Web Link: RC Circuit II
q  q0e
time

t
RC
Recall: Electric Field (E) points from + to - charge
Magnetic Field (B)
points from “North” to “South” poles
opposite poles attract
like poles repel
Magnetic Field Lines
B is tangent to the field
lines at any point
The density of the
lines represents the
strength of the
magnetic field
Web Links: Magnetic Field
3-D Magnetic Field
Facts about Magnetic Fields (B-fields)
1) North and South poles cannot be isolated
2) All B-fields are caused by moving electric charge
3) The Earth has a Magnetic Field:
Web Links: Northern Lights
4) B-fields exert a force on moving, charged particles:
+ Force is out of the screen
unaffected +
+ unaffected
B
+ Force is into of the screen
Magnetic Force = F = qvBsin
q = charge
v = speed of charge
What is the direction of this force?
Right Hand Rule (RHR)
(For a positive charge)
point with v
Then curl toward B
Thumb points with F
B = magnetic field
 = angle between
Fingers
v and B
F
B


SI unit for B-field
is a Tesla (T)
Other unit:
1 Gauss = 10-4 T

v
(F is in opposite direction
for a negative charge)
Since it’s difficult to draw in 3-D, we’ll adopt the
following symbols:
dots indicate a B-field
out of the page
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x’s indicate a B-field
into the page
(hint: just think of arrows:
)
Web Links: Charged particles in a Magnetic Field
Deflection of a moving electron
In the following examples, is the charge + or - ?
x
x
x
x
?
x
x
x
x
x
x
x
x
x
x
x
x
?
?
Work done by the Magnetic Force
x
x
x
x
x
x
x
F
F
+
v
x
x
F
s
s
x
x
x
s
x
x
x
x
Work = (Fcos)s = ?
The work done by the Magnetic Force is equal to _____
The speed of a charge in a Magnetic Field is ______
Circulating Charged Particle
When the charge moves
perpendicular to the B-field,
we can show that:
mv
radius  r 
qB
2 m
period  T 
qB
qB
frequency  f 
2 m
Web Link: Charge in 2 Magnetic Fields
What path does the charge follow
if v is not perpendicular to B?
Web Link: Helix
Ex:
-
An electron in a magnetic field moves at a speed
of 1.3 x 106 m/s in a circle of radius .35 m. Find
the magnitude and direction of the magnetic field.
Crossed () Electric and Magnetic Fields
B
-
v
x
x
x
x
x
x
x
x
x
x
E
As the electron enters the crossed fields:
The Electric Field deflects it in what direction?
The Magnetic Field deflects it in what direction?
If E and B are adjusted so that the electron continues
in a straight line…
E
v
B
Web Links: Magnetism inside a TV, TV Screens
Another example of Magnetic and Electric fields
working together: A Particle Accelerator
The Large Hadron Collider (LHC), on the border of
France and Switzerland, has a circumference of 16.7
miles. It accelerates particles to near the speed of light,
so that high energy collisions can be used to further
study the structure of matter. (Web Link: LHC News)
What happens to a current-carrying wire in a B-field?
Remember: current is just
moving charge
B
L
I
F = I L B sin
What is the direction of
force on this wire?
We can derive an
equation for the
magnitude of this force…
 = angle between
B and current
Ex:
x
x x x
x
B = .440 T
x
x x x
x
L = 62.0 cm
x
x x x
x
m = 13.0 g
x x x x x x
x
L
Find the magnitude and direction of the current
that must flow through the red bar in order to
remove the tension from the springs.
Make sure you
don’t confuse these
two separate effects:
1) A Magnetic Field exerts a force on a Current
2) A Current produces its own Magnetic Field
Magnetic Field due to a long straight current:
B
Right Hand Rule #2
Thumb points with I
Fingers curl with B
I
The magnitude of B
depends on the distance r
from the current:
r
0 I
B
2 r
0 = 4 x 10-7 Tm/A
permeability of free space
Weblink:
Right Hand Rule
Ex:
If a wire carries a current of 480 A, how far from
the wire will the magnetic field have a value of
5.0 x 10-5 T ?
(roughly the value of earth’s magnetic field)
Parallel Currents
d
B1
x
x
x
x
I1
I2
Current I1 produces a
B-field
L
This B-field
exerts a force
on current I2
(and vice versa)
What is the direction of force on I2 due to I1 ?
(hint: use both right hand rules)
What is the magnitude of force on I2 due to I1 ?
(hint: use both equations)
Consider a circular current…
and use RHR #2 to determine the direction of
the magnetic field at the center of the loop:
I
B
I
B
B
B
B
x
or
B
B
I
I
At the center of the loop:
B
 0I
2R
Radius of loop
II
I
If there are many circular loops:
BN
0 I
2R
N = number of loops
Web Link: Compass in loops of current
Magnetic Fields add as vectors
I
I
I
At the center of the loop:
The straight section creates a B-field
The circular section creates a B-field
Do these
fields add or
subtract?
I
I
I
Do the B-fields add or subtract in this case?
Solenoid
inside:
xxxxxxxxxxxx
I
B
I
For a long, ideal solenoid: B = 0n I
n = turns/length
Web Link: Solenoid Factors
What are solenoids used for?
car starters
doorbells
Web Link:
How doorbells work
electric
door
locks
Ex:
20 cm
The solenoid has 100 turns. If a current of
23 A runs through it, what is the magnitude
of the magnetic field in its core?
Toroid
In video games, what
does it mean to play in a
“toroidal world”
Web Link: Asteroids
Asteroids
Magnetic Flux ()
is related to the number of magnetic field
lines passing through a surface
B
B
S
N
B
From above
Web Link: Flux
Magnetic Flux =  = B A cos 
SI unit =
1 Weber
= T·m2
B = magnetic field
A = surface area
 = angle between B and the
Normal to the surface
Ex:
square
loop
2.0 m
B = 5.0 x 10-4 T
a) What is the angle  in this example?
b) Calculate the magnetic flux through the loop
c) What happens to the flux if the normal is rotated
by 30° ?
d) What happens to the flux if the normal is rotated
by 90° ?
Recall: An emf is anything that produces
a voltage difference (and therefore
causes current flow)
Recall: For a current loop, we can
determine the direction of
the B-field at its center:
I
B
Here’s a quicker
way to do this:
B
I
I
I
Loop Right Hand Rule
Fingers curl with I
Thumb points with B
B
x
I
Faraday’s Law of Electromagnetic Induction
An emf is induced in a conducting loop whenever
the magnetic flux () is changing.

emf  
t
Web Links: Induction, Faraday’s Experiment
Notes: 1) /t = rate of change of flux
2) Induced emf causes induced current in the loop
3) Induced current causes its own magnetic field
4) This new B-field direction opposes the change in
the original one. This part is called Lenz’s Law.
Web Link: Lenz’s Law
5) If there are multiple loops:

emf   N
t
(N = number of turns)
B
A
Here is a conducting loop in a magnetic field
Magnetic Flux =  = B A cos 
Can you think of 3 different ways
to induce a current in this loop?
Ex:
B
S
N
As the loop moves to the left, what is the
direction of the current that is induced in it?
As loop moves left:
Ex:
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
As the loop is pulled and its area is
decreased, what is the direction of the
current that is induced in it?
Web Link: Induced current
Notice in the previous examples:
If the magnetic flux is increasing, the induced B-field
is in the opposite direction as the original B-field
B
If the magnetic flux is decreasing, the induced B-field
is in the same direction as the original B-field
B
Web Link: Lenz’s Law
Ex:
B
N
S
Find the direction of current in the loop when:
a) The magnet moves to the left
b) The loop moves to the left
c) Both the magnet and loop are stationary
Ex:
20 cm
20 cm
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
B = 2.0 T
The wire loop has a resistance of 20 m. If its
area is reduced to zero in a time of .20 s, find the
magnitude and direction of the induced current.
Finally…
why does it take so long for a magnet to
fall through an aluminum pipe??
Web Link: Lenz’s Law Pipe
There are many familiar
examples of induction
all around us…
Generator
Web Link: Generator
Dynamic Microphone
Web Link: Dynamic Microphone
Speakers
Web Link: How a speaker works
Electric Guitar
Web Link:
Electric Guitar
Motional emf
x
x
x
x
L
x
x
x
x
conductor
B
x
x
x
speed v
x
x
What happens to
the positive charge
on the conductor?
x
x
x
What about the
negative charge?
Potential difference
between the top and
bottom =
Motional emf = vBL
Ex:
If the conducting bar is moved along conducting
rails as shown below, we can see that there will
be a current in the direction indicated:
Could we have found the current direction using
Lenz’s Law instead?
Near San Francisco,
where the vertically
downward component
of the earth’s magnetic
field is 4.8 x 10-5 T, a
car is traveling forward
at 25 m/s. An emf of
2.4 x 10-3 V is induced
between the sides of
the car.
a) Which side of the car is positive, the driver’s or
passenger’s?
b) What is the width of the car?
Circuits
DC voltage source
AC voltage source
Resistor
Capacitor
E-field inside
Inductor
(Solenoid)
B-field inside

If N = number of turns
I = current
 = magnetic flux
N
Inductance = L 
I
SI unit = Henry(H) = Wb/A
The inductance (L) of a solenoid is not determined by
the current or flux through it at a particular moment.
L is a fixed property of each inductor:
A

L = 0 n2 A ℓ
Recall:
n = turns / length
Inductors store energy in their B-fields:
Energy stored in an inductor = ½ L I2
Energy
B2
Energy Density 

Volume 20
How do inductors behave in circuits?
L
B
Constant I
I
I
+
-
Changing I
Constant B
very boring
Changing B
Induced emf
Since there is only one
inductor, this is called
Self-Induction
Changing 
I
emf   L
t
voltage
across
inductor
Opposes
change in I
When two inductors affect each other, it is called
Mutual-Induction
1
2
2
B1
N2 turns
I1
If I1 changes
B1 changes
2 changes
+
-
Mutual Inductance =
N2  2
M
I1
emf2 induced
in circuit 2
 I1
emf2   M
t
Secondary
Circuit
Primary
Circuit
During a 72-ms interval, a change in the current in
a primary coil occurs. This change leads to the
appearance of a 6.0-mA current in a nearby
secondary coil The secondary coil is part of a circuit
in which the resistance is 12 . The mutual
inductance between the two coils is 3.2 mH. What
is the change in the primary current?
Recall : Power = I V
Current is reduced to
minimize power loss
V
I
Voltage is reduced to
household levels
IV
How is the power line voltage raised
and lowered?
Transformer Station
Transformer increases (steps up) or decreases (steps down)
ac voltage using induction
Web Link: Faraday’s Transformer
Transformer:
Iron
generator
Primary Coil
Voltage VP
NP turns
Web Link:
Transformer
Secondary Coil
Voltage Vs
NS turns
VS NS

VP NP
Transformer
Equation
Ex:
120 V
3.0 A
?
Find the output voltage and current.
Recall the difference between AC and DC:
Web Link: AC vs. DC
DC (
):
Ex:
Voltage
time
AC (
):
V0
Voltage
Ex:
time
-V0
V = V0 sin ( 2  f t )
Voltage amplitude
frequency
time
Before we study AC circuits, let’s prepare by reviewing
how the circuit components behave in a DC circuit:
I
V
I = V/R
R
R
I
V

C
R


I
V
L

I = V/R at the first instant,
then it decreases until I = 0
At this point, the capacitor
is fully charged, and acts
like a break in the circuit
Induced emf across L slows
current increase until I = V/R
At this point the flux is no
longer changing, and the
inductor acts like a wire.
Resistor in an AC Circuit
V=
V0sin(2ft)
R
Vrms 
Irms 
What about the
instantaneous values?
Irms
Web Link: AC Circuits
V
I
t
t
V0
2
I0
2
These are all
average
values
Vrms

R
Voltage and Current are
in phase in a purely
resistive circuit.
Capacitor in an AC Circuit
Acts like a resistor:
Vrms
C
f
Irms
Vrms

XC


1
R = XC 
2 f C
Capacitive Reactance
SI unit = Ohms ()
What happens to XC when the
frequency is very large ??
What happens to XC when the
frequency is very small ??
Instantaneous Values for
a Capacitor in an AC Circuit
Web Link: AC Circuits
Capacitor is full here:
q=0
Capacitor is charging
fastest when empty
V
t
I
(q/t)
t
Current leads Voltage
by 90° in a purely
capacitive AC circuit
Power = I V
one is maximum when
the other is zero
Average Power ( P ) = 0 for a capacitor in an AC circuit
Inductor in an AC Circuit
Acts like a resistor:
L
R=
Irms
Vrms

XL


XL  2  f L
Inductive Reactance
SI unit = Ohms ()
What happens to XL when the
frequency is very small ??
What happens to XL when the
frequency is very large ??
L
Instantaneous Values for
an Inductor in an AC Circuit
I is not changing:
V=0
I
Web Link: AC Circuits
I decreasing fastest:
V is minimum
I increasing fastest: V
is maximum
t
V
( I/t)
t
Current lags Voltage by
90° in a purely
inductive AC circuit
Power = I V
one is maximum when
the other is zero
Average Power ( P ) = 0 for an inductor in an AC circuit
Series RCL Circuits
Acts like a resistor:
R =Z
R  XL  X c 
2
Impedance ()
Irms
Vrms

Z
 XL  X C 
Phase Angle between I & V =  = tan 



R
1
Average Power ( P ) = Irms Vrms cos 
cos  = power factor
2
Ex:
16.0 
15.0 V
1350 Hz
4.10 F
5.30 mH
a) Find Irms
b) Find the voltage across each circuit element
c) Find the average power dissipated in the circuit
Non-Series RCL Circuits
a) Find Irms for a very
large frequency
b) Find Irms for a very
small frequency
Vrms , f
Resonance in AC Circuits
Oscillating systems:
Mass on a spring
PE
AC Circuit
E-field
KE
PE
I
++++
B-field
----
I
Web Link:
Electromagnetic
Oscillating Circuit
C
L
This circuit has a
natural frequency
1
f0 
2 LC
Resonant frequency for an RCL circuit
(independent of R)
Ex: Tuning a Radio
Web Link: Radio Tuning
Electromagnetic Wave
Mutually perpendicular and oscillating
Electric and Magnetic fields
Web Link: Electromagnetic Wave


Electromagnetic waves are transverse waves
Electromagnetic waves travel at the speed of light
in a vacuum: c = 3.00 x 108 m/s
Recall these facts:
1) A changing B-field produces an E-field
+
atom
-
E-field
B
2) A changing E-field produces a B-field
B-field
E-field
B-field
It could go on forever!
This is how to make an electromagnetic wave
Web Links: Propagation of an electromagnetic wave
Vibrating Charges
The Electromagnetic (e/m) Spectrum
c=f
speed of light
wavelength
frequency
Web Link:
Wavelengths
Remember these constants?
0= permittivity of free space
0= permeability of free space
Fundamental
constants of
nature
In 1865, Scottish physicist James Clerk
Maxwell hypothesized electromagnetic waves
and calculated that they would have to travel
at a specific speed in a vacuum:
1
Do the calculation.
 o 0
What do you get?
This is the measured speed of light!
Electromagnetic Waves do exist,
and light must be one of them!
Our Reference Frame determines
where and when we observe an event:
y
y
x
x
z
z
const. velocity
In both cases, the Reference Frame is
at rest with respect to the observer
For each of the cases below, what path
does the observer see the ball follow after
he throws it straight up?
on the
ground
in a truck with
constant velocity
Inertial Reference Frames
(constant velocity)
in a truck with
constant acceleration
Non-Inertial
Reference Frame
Special Relativity Postulates
1) The laws of physics are the same
in any inertial reference frame.
2) The speed of light in a vacuum (c) has the same
value when measured in any inertial reference frame,
even if the light source is moving relative to it.
speed
of light
speed
of truck
Result

For speeds far less than c, relativity is barely noticeable

For greater speeds, observers in different
reference frames experience:
a) Time Dilation (time slows down)
b) Length Contraction (things shrink)
Time Dilation
Imagine a
“light clock”
Now imagine putting
it on a spaceship.
To an observer on the
ground, what path
does the light follow?
t
 t0
2
v
1
c2
Time
Dilation
Equation
t0 = proper time (measured in the same reference
frame as the events are occurring)
t = time measured by an observer in a different
reference frame
v = relative speed between the two reference frames
c = 3.00 x 108 m/s
So what does this all mean ???
t
 t0
1 v
<1
2
c
t > t0
2
<1
Time slows down in a reference frame
that is moving relative to the observer !
Web Link: Time Dilation
Proof:
1) Atomic clocks on jets slow by
precisely this amount
2) GPS and airplane navigation
must use it in their calculations!
3) Muons arrive at earth’s surface
Web Link: Muon Time Dilation
Ex:
An observer on the ground is
monitoring an astronaut in a
spacecraft that is traveling at a
speed of 5 x 107 m/s .
On average, a human heart beats 70 times per
minute. Calculate the time between heartbeats
and the number of heartbeats per day for
a) the person on earth (this part is easy)
b) the space traveler, as monitored from earth
So the guy on the
ground sees the guy
on the spaceship
aging more slowly.
What does the guy
on the spaceship see
when he looks at the
guy on the ground ??
The Twin Paradox
One twin travels at a speed of .80c to a
galaxy 8 light years away and and then
travels back to earth at the same speed.
Upon his return he will be 8 years
younger than his twin!
How is this different from the previous example ??
Understanding Time Dilation
y
More y-motion, less x-motion
Constant speed in x-direction
x
space
More motion through space,
less motion through time
Sitting still
(not moving through space)
time
Just think of time as the 4th dimension
Length Contraction
Observer (t)
(t0)
v = relative speed
v
L0
L0 = proper length (measured by observer at
rest with respect to object/distance)
L = length measured from a different reference
frame
c = 3.00 x 108 m/s
L  L0 1  v
2
c2
Length
Contraction
Equation
<1
Distances/lengths appear shorter when
moving relative to the observer.
Web Link: Length Contraction
*Only in the direction of motion:
v
Ex: Passing spaceships
spaceship 1
(2.0 x 108 m/s)
spaceship 2
(at rest)
Both have a proper length of 8.5 m.
How long does spaceship 1 look to spaceship 2 ?
How long does spaceship 2 look to spaceship 1 ?
Recall:
m1
momentum = p = mv
v1
v2
m2
Conservation of Momentum:
m1v1 + m2v2 = constant
When things are moving close to the speed of light,
this equation is way off !
We need to consider…
Relativistic
Momentum
p
mv
1 v
>mv
2
c2
<1



If we calculate momentum this way
for high speeds, conservation of
momentum is obeyed.
What happens if we use this equation when
v is very small ?
Are there any situations in which things move
so fast that we have to use this equation?
Stanford Linear Particle Accelerator
Electrons accelerate to 99.99999997% speed of light !
Momentum is 40,000 times greater than mv !
E = mc2
Mass-Energy Equivalence
Mass
Energy
conserved together
Total Energy of an Object = E 
If v=0 :
This much
energy
mc 2
2
v
1
E0 = mc2
= rest energy
is
equivalent
to
This much
mass
c2
E0 = mc2
A huge
amount of
energy
A small
mass
The rest energy of a 46 gram golf ball
could be used to operate a 75-Watt
light bulb for 1.7 million years!
Ex:
Our country uses about 3.3 trillion kWhrs of energy
per year. Find the amount of mass that is
equivalent to this much energy.
E0 = mc2
If energy
changes
Mass must
change also
Why don’t we notice this ?
When a 1 kg ball falls 200 m and
lands on the ground, by how
much does its mass change?
More examples of Mass-Energy Equivalence…
Ex: Matter meets antimatter
eelectron
+
e+
=
positron
2 (9.11x10-31 kg)
People used to wonder if
the moon was made of
antimatter
gamma rays
mass = 0
pure energy
Ex: Nuclear Power (Fission)
Big nucleus
2 smaller nuclei
(less total mass, less energy)
Web Link: Fission
Ex: The Sun (Fusion)
Two small nuclei
Larger nucleus
(less total mass, less energy)
Web Link: Fusion
The sun loses over 4 billion kg
per second due to fusion
(Don’t worry, it will last for another 5 billion years or so)
Recall:
E0 = mc2 = rest energy
If an object is moving, its total energy is the sum
of its rest energy and its kinetic energy:
E = E0 + KE




We can solve
1
Relativistic
2
KE  mc 
 1
for KE…
2
 1 v 2
 Kinetic Energy


c
What happens to this equation if an
object is traveling at the speed of light?
Objects with mass cannot
reach the speed of light
Recall that all these effects of Special Relativity
would only become noticeable to us as speeds
approach the speed of light.
Let’s try to get an idea of how fast light really is…
Traveling at the speed of light, just
how far around the earth could you
go in 1 second?
When they are headed for the same place
at the same time…
Particles experience:
Waves experience:
Collisions
Interference
Electrons are…
Particles:
-
Interference
and Waves:
Web Links:
Electron Interference
Double Slit Experiment
Light is…
a Wave:
and a Particle:
light
collisions
Wave-Particle Duality
metal
Photoelectric
Effect
Light (any electromagnetic wave) is composed of …
Photons – massless energy particles
E=hf
E = Energy of 1 photon
h = Planck’s constant
= 6.626 x 10-34 Js
f = frequency of light wave
Ex:
How many photons are emitted in 1 hour by a
25 Watt red light bulb ? ( For red, use =750 nm)
Ex:
Which type of electromagnetic wave is represented
by photons with the following energies ?
a)
E = 3.3 x 10-16 J
b)
E = 1.3 x 10-20 J
The Photoelectric Effect
Web Link: Photoelectric Effect
Photon
E=hf
Electron with
maximum KE
Conservation of Energy:
W0 = Work Function
= minimum work
required to eject an
electron from the metal
hf = W0 + KEmax

No electrons are ejected if the frequency is too low

More light does not result in electrons with more KE
Energy is being absorbed in packets (like particles)
The Photoelectric Effect in the garage…
More Photoelectric Effect Applications
Photographer’s
light meter
Automatic Doors
Web Link: Solar Energy
Digital Camera
Web Link:
Digital Camera
Ex:
White Light (all colors)
 = 380-750 nm
-
Sodium
(W0=2.28 eV)
-
Find the maximum kinetic energy of the
ejected electrons (in electron-Volts).
The Compton Effect
Web Link: Compton Effect
(Energy=hf)
(Energy=hf’)
The electron
now has some
Kinetic Energy
Does the photon
have more or
less energy after
the collision?
Photon Momentum  p 
h

e
e


Conservation of Energy &
Conservation of Momentum…
h
1  cos 
   
mc
’
h = Planck’s constant
m = electron mass
c = speed of light
 h 

 = Compton wavelength = 2.43 x 10-12 m
 mc 
What is the change in wavelength if =0°? =180°?
Now take a few minutes to discuss
these with your group:
Conceptual Example 4
in the textbook (p.905)
Solar Sail
Check Your Understanding
#10
(p.906)
Radiometer
OK, so we’ve accepted the fact
that waves act like particles
(have momentum, collisions, etc.)
p
h

In 1923 Prince Louis de Broglie
suggested for the first time that maybe
particles act like waves:
h
De Broglie Wavelength  
p
When they finally tried it out
with electrons, the interference
pattern corresponded perfectly
to this wavelength!
Ex:
Find the de Broglie wavelength of a car
with a mass of 1000 kg traveling at a
speed of 30 m/s.
So what does this wavelength
really mean for particles??
It’s a Probability Wave:
100 electrons
70000 electrons
3000 electrons
Web Link:
Does the universe exist
if we’re not looking???
The Heisenberg Uncertainty Principle
“The more precisely the position
is determined, the less precisely
the momentum is known”
- Heisenberg, Uncertainty paper, 1927
If
x = uncertainty in position,
and p = uncertainty in momentum,
then
h
  x p 
4
Ex:
Within an atom, the uncertainty in an electron’s
position is 10-10 m (the size of the atom).
Find the uncertainty in the electron’s speed.
Ex:
10 cm
The marble (m=25 g) is somewhere within the
box. Find the uncertainty in the marble’s speed.
Heisenberg is out for a drive when he’s
stopped by a traffic cop. The cop says “Do you
know how fast you were going?”
Heisenberg says “No, but I know where I am.”
There is another form of Heisenberg’s Uncertainty
Principle that involves Energy and Time:
If
E = uncertainty in a particle’s energy,
and t = the time it has that energy,
then
h
  E  t 
4
This leads to “Quantum Tunneling”
Web Links: Scanning Tunneling Microscope
Animated STM
STM images
The best part about knowing all
this physics, is that now you
will get the jokes……
A Party of Famous Physicists
Let’s see how many of the following
physicists you can guess…
Everyone was attracted to his
magnetic personality.
He was under too much pressure to
enjoy himself.
?
?
?
He may or may not have been there.
He went back to the buffet table
several times a minute.
He turned out to be a powerful
speaker.
He got a real charge out of the whole
thing.
He thought it was a relatively good
time.