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BELLRINGER
Compare and explain in
complete sentences and formulas
what is the unit for nuclear force.
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Homework due tomorrow
WHAT IS THE LAW OF
CONSERVATION OF ENERGY?
GIVE EXAMPLES.
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There are Four Fundamental Forces:
1) The Electromagnetic Force
3) The Strong Nuclear Force
4) The Weak Nuclear Force
(We’ll study it this term)
These act over a very small range
These are responsible for
all we see accelerate 2) The Gravitational Force
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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!
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Now let’s review the gravitational force…
Any two masses are attracted by equal and opposite gravitational forces:
m1 m2
r
F -F
Newton’s Universal Law of Gravitationwhere…… F G
m mr1 2
2
G=Universal Gravitation Constant = 6.67x10-11 Nm2/kg2
This is an Inverse-Square force Gravity is a very weak force
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atom
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
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silk
glass
(rub)
- --
(rub)
- --
+ + + +
fur
plastic- - - -
Web Links: Static DusterNew Carpet
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Ex:
If you rub a balloon against your hair, which ends up with more electrons, the
balloon or your hair?
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Opposites Charges Attract
Like Charges
Repel
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Insulators (like plastic, rubber, pure water, and glass) will
not conduct away extra charge.
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
Use rubber gloves in the lab
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Grounding
- - - -
The earth is a huge reservoir of positive and negative charge
+ +
+
+
++
+
+
+-
-
--
-
--
--
-
-
---
- Object is discharged or “grounded”
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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 1Charging by Induction 2
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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
I
I q t Amperes (A) Coulombs (C) seconds (s)
SI units
q t
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Remember, opposite charges attract:
and like charges repel:
q1 and q2 may
represent lots of extra or missing
electrons
How much force do q1 and q2 exert on each other?
Coulomb’s LawF electrostatic force k
q q
r1 2
2
k = electrostatic constant = 8.99 x 109 Nm2/C2
Web Link: Orbiting electron
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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:
k1
4
00= 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
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Ex:
+q +q-q
r 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?
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Ex:
q1
q2
q3
.15 m
.10 m
73°
q1= +4.0 C
q2= -6.0 C
q3= -5.0 C
Find the net force on charge q1
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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:
q = ne , where n = 1, 2, 3, etc…
…
Charge is quantized Also:
Charge is conserved
= e = elementary
charge
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Ex:
- +
electron proton
1.0 cm
Calculate both the gravitational force and the electrostatic force, and compare their magnitudes.
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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
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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
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Electric Field (E) – A vector field surrounding a fixed, charged object that indicates the force on
a positive test charge (q0) placed nearby
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.
fixed, charged object
+
++ +
+ ++
+
+
+ +test charge
Web Link: Force Fields
EF
q0
The Electric Field is defined as the Force per unit Charge at that point
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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:
EF
q0
F = q0E
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Ex:
+
q = 2.0 C
(fixed charge)
+
q0 = 1.0 C (test 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?
.10 m
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Ek q
r 2
Electric Field at a distance r from a point charge q
r
E = ?
q
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+
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?
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What do you think the Electric Field lines would look like for…
A charged, non-conducting sheet that is not infinite?
+
++ +
+ +
+
+
+
+
A large (), charged, non-conducting sheet?
+
++ +
+ ++
+
+
++
+
++ +
+ +
+
+
+
+
--
--
-
-
Two oppositely charged plates?
(called a parallel plate capacitor)
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The Electric Field Lines for 2 Equal Charges:
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The Electric Field Lines for 2 Opposite Charges (called an Electric Dipole):
Web Links: Electric Field Lines, Releasing a test charge
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Charged Conductors
Any excess charge ends up on the surface of a conductor, independent of its shape
Why do you think this happens?
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What happens to a neutral conductor placed in an external electric field?
“Shielding”
E = 0
At equilibrium, the Electric Field at any point within a
conducting material is zero.
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Faraday Cage: an example of shielding
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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) f)
d)
e)
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Recall…
Gravitational Potential Energy
or
Elastic Potential Energy
Now…
-+
+++++
+
++
+ +
Electric Potential Energy
(EPE)
Only Conservative Forces have an associated PE
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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
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Ex:
Uniform Electric Field
+proton
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.
+
2.0 m
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Work is Path Independent for conservative forces:
path 1
path 2
Work done by gravity on path 1 = Work done by gravity
on path 2
Ex: Electric Field
path 1
path 2
Work done by electrostatic force
on path 1 =
Work done by electrostatic force
on path 2
Ex: Gravity
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EPE is a type of mechanical energy, like…
Kinetic Energy (KE) = ½ mv2
Rotational Kinetic Energy (KER) = ½ I2
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).
+++
+
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Ex:
Uniform Electric FieldE = 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 ?
+proton
Can you think of two different methods to use in solving this problem?
Do they yield the same answers?
+
1.0 m
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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:
Electric Potential VEPEq0
= EPE per charge
Also called Potential or Voltage
SI Unit = J/C = 1 Volt
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From the definition of Electric Potential, we can show that when a charge is moved from
one point to another in an electric field:
Work done by the Electric
Field= -
Charge that was moved
Difference in Potential between its old and
new positions
E1
2
W = -q0(V)
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Let’s make sure that we understand the difference between Potential and Electric Potential
Energy:
V (in Volts) = Potential
a property of a certain position in an Electric Field with or without
charges placed there
E
-
EPE (in Joules) = Electric Potential Energy
a property of charges placed at a certain position in an external Electric Field +
- EWeb Link: EPE vs Potential
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We now have a new SI unit for Electric Field:
Volts / meter
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
E = 3 N/C = 3 V/m
Ex
-
1 V
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Equipotential Surfaces adjacent points at the same electric potential
E-field Equipotential Surface
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Web Link: Equipotential surfaces
Equipotential Surface
E-Field
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Equipotential Surfaces are 3-dimensional:
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Equipotential Surface
E-Field
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:
+
s
F
Web Link: Equipotential surfaces
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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)
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Ex:
.30 m
E = 5.0 V/m
Find the potential difference between the plates.
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In the lab, we could use a Voltmeter to simply measure the potential difference:
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This means there is a potential difference (V) of 12 Volts between the terminals of the battery
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Calculating the Potential due to a Point Charge
q
r
What is the Potential at this point?
V kqr
k = electrostatic constant = 8.99 x 109 Nm2/C2
Notes:
1) Include the sign of q in your calculation! (+ or -)
3) The equation can also be used for a charged sphere:
+
++++ +
++
++
rV k
qr
Total charge
Distance from center
2) Potential Difference can also be calculated:
V = V2 – V1 k
qr
kqr2 1
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Van de Graff generator
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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?
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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 = V1 + V2 + V3 + … (an
algebraic sum, not a vector sum)
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Ex:+q
+q
-q -qP
Find the potential V at point P due to the four charges.
dd
d
d
Web Link: Complex Electric Field
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Capacitor a device that stores energy by
maintaining a separation between positive and negative charge
(Symbol: )
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Circuit Board
Capacitor
Resistors
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Parallel Plate Capacitor
--
+q
-q
This is called “charging a capacitor”
q = charge of the capacitor
V
V = potential difference of the capacitor
q and V are proportional:
q = C V
C = Capacitance (a fixed property of each capacitor)
SI unit = 1 Farad (F) = 1 Coulomb / Volt
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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
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Calculating the Capacitance (C) of a parallel plate capacitor
A
A = plate area
d
d = plate separation
= dielectric constant
CA
d 0
(0= 8.85 x 10-12 C2/Nm2)Notice:
Capacitance is independent of both charge and voltage
Adding a dielectric increases the Capacitance
Web Links: Capacitance Factors, Lightning
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How much Energy is stored by a capacitor?
Energy = ½CV2
VoltageCapacitance
What’s the energy density in an Electric Field?
Energy DensityEnergyVolum e
E 1
2 02
* For any electric field
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+q
d
-q +q
D
-q
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
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Here’s a Web Link about a huge capacitor and
what can be done with all that stored energy:
Pulse Discharge Machine
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Imagine a wire:
V
+-
E
Web Link: DC Electricity
Now imagine bending the same wire into a loop: +
-V
- - -
Battery or other emf source
emf – electromotive “force” – the potential difference between the terminals of an electric power source
-
-
-
Ex:
emf = 9 V
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+
-+
The current arrow points with the “positive charge carriers”
current Iqt
Web Link: Conventional Current
Notes on Current:
1) Remember: charge is conserved
SI unit = Ampere(A) = 1 C/s
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
I
+
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Ex:
I
A DC current of 5.0 A flows through this wire:
How much charge flows past this point in 4.0 minutes?
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Will the bird on the high voltage wire be shocked?
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Resistance RVI
applied voltage
resulting current
SI unit: Ohm () = 1 V/A
(Resistor symbol: )
Resistor – a circuit component designed to provide a specific amount of resistance to current flow.
Web Link: Resistance
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9 V1000
Ex:
Draw the circuit diagram, and calculate the current in this circuit.
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Building Resistors
Resistance = R = a property of a given resistor (Ex: 20 , 400 , etc.)
Resistivity = = a property of a material used in making resistors
A
L
RLA
(: SI unit = ·m)
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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.
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Ex: Incandescent Light Bulb
120 V
I = 12.4 A
Tungsten wire radius .045 mm
What is the length of the tungsten wire inside the light bulb?
Web Link: Light bulb
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V = I R( I V )
I
V
( I V )
Is it really a law ?“Ohm’s Law”
It works for resistors:
What about other devices?
Diode
I
V
Light Bulb
I
V
“Ohm’s Law” is not really
a Law!
( I V )
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Power = P = IV
SI Unit = 1 Watt (W) = 1 J/s
Rate of energy transfer
If the device is a resistor:
P = I V
V=IR
= I2R
P = I V
I=V/R
= V2/R Energy dissipated by the resistor as thermal energy
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Ex: Space Heater
1500 W Heater
120 V
Find:a) The resistance of the heaterb) The current through the heaterc) The amount of heat produced in 1 hour
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…back to the difference between AC and DC:
Web Link: AC vs. DC
DC ( ) :
Voltage
time
Ex:
AC ( ) :
Voltage time
Ex:
V = V0 sin ( 2 f t )
Voltage amplitude frequency time
radians
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So what does AC current look like?
Typical household
outlet:V0 = 170 V f = 60 Hz
Light bulb: Resistance R
IVR
V sin 2 f t
R0
= I0 = current amplitude
I = I0 sin ( 2 f t )
It
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Ex: Alarm Clock
How many times a day does the current change direction?
V0 = 170 V f = 60 Hz
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look familiar??
AC PowerP = I V = ?
II
2rm s0 V
V
2rm s0
peak values
These are the values that matter
P = Irms Vrms
P = (Irms)2 R
P = (Vrms)2 / R
Ex:
V0 = 170 V
What is the rms voltage?
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Ex: Speaker
If the power rating of the speaker is 55 Watts, and its resistance is 4.0 ,
what is the peak voltage?
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Heating element of resistance R
AC generator
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Resistors in Series
R1 R2 RS = R1 + R2
(RS > R1 , R2)
Resistors in Parallel
R1
R2
1R
1R
1RP 1 2
(RP < R1 , R2)
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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.
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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
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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 ?
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Basic Circuit: RV I = V/RI
IR1
Series Circuit:
V
R2
Current (I) has the same value everywhere in the circuit
current is like a parade
VR1 + VR2 = VBattery
voltage is like money
RSV
I I = V/RS
RS = R1 + R2
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RPV
I1
Parallel Circuit:
I2 R1V R2
I3
I1
?
I1 = I2 + I3
VBatt = VR1 = VR2
Web Link: Parallel Current
1R
1R
1RP 1 2
I1 = V/RP I2 = V/R1
I3 = V/R2
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Ex:
416 V
4
What is the series resistance?
Calculate the current in this circuit.
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16 V 4 4
What is the parallel resistance?
Calculate the current in all branches of this circuit.
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47 V
28
Ex:
The current through the 47 resistor is .12 A Calculate the voltage V of the battery.
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Ex:
V 47 28
The current through the 47 resistor is .12 A Calculate the current through the 28 resistor.
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R1
V
R2
In a series circuit, the current is the
same through each resistor
R1V R2
In a parallel circuit, the voltage is the same across each resistor
Notice that the terminology will help us remember how to measure current and voltage
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Measure the voltage across a resistor:
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Measure the current through a resistor:
You must break the circuit to
measure current!
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How to calculate the equivalent resistance for a group of resistors:
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Ex:
Find the equivalent
resistance of this circuit:
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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.
Web Link: Kirchoff’s 1st Law
I1
I2
I3
I4
I1+ I2+ I3= I4
Ex:
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II) The Loop Rule
The potential differences around any closed loop sum to zero.
Web Link: Kirchoff’s 2nd Law
I2
I1
I3+
-
+
-
+
-+
-
V = I R
VR1 = I2R1
VR2 = I2R2
VR3 = ?
+V - I2R1 - I2R2 = 0
This loop (clockwise): Write out the equations for this loop and the outer loop
Ex:
R1
VR2
R3
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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
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Ex:
8.0 V V
3.0
4.0
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
1.7 A
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Capacitors in Circuits
CA
d 0A
d
Recall:C A
C 1/d
C1V C2
Capacitors in Parallel:
CP = C1 + C2
Capacitors in Series:
C1V
C2
1C
1C
1CS 1 2
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Ex:
8.0 F 5 V
6.0 F
4.0 F
a) Find the total capacitance of the circuit
b) Find the total charge stored on the capacitors
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Charging a Capacitor:Web Link: RC Circuit I
At t = 0: close the switchFirst instant: I = V0/RThen: I decreases as the capacitor fills with charge
Finally: I = 0, and Vcap = Vbattery = V0
Web Link: RC Circuit II
Charg
e o
n
cap
aci
tor
time
q0 = CV0
full capacitor charge
RC = time constant =
q q 1 e0
tRC
RC Circuits
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Discharging a Capacitor:
At t = 0: close the switch
First instant: I = V0/R
Then: I decreases as the capacitor loses its charge
Finally: I = 0, and Vcap = 0
Web Link: RC Circuit I
The capacitor starts out fully charged to voltage V0
Charg
e o
n
cap
aci
tor
time
Web Link: RC Circuit II
q q e0
tRC
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Magnetic Field (B) points from “North” to “South” poles
Recall: Electric Field (E) points from + to - charge
opposite poles attract like poles repel
Magnetic Field LinesB 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
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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
B + Force is into of the screen
+unaffected +unaffected
+
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Magnetic Force = F = qvBsin
v = speed of charge
B = magnetic field
= angle between v and B
q = chargeWhat is the direction of this force?
Fingers point with vThen curl toward BThumb points with F
SI unit for B-field is a Tesla (T)
(F is in opposite direction for a negative charge)
Other unit: 1 Gauss = 10-4 T
Right Hand Rule (RHR) (For a positive charge)
B
v
F
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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 xx x x xx 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
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In the following examples, is the charge + or - ?
x x x x
x x x x
x x x x
x x x x
?
??
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Work done by the Magnetic Force
x x x x
x x x x
x x x x
x x x x+
v
s
s
s
F
F
F
Work = (Fcos)s = ?
The work done by the Magnetic Force is equal to _____
The speed of a charge in a Magnetic Field is ______
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Circulating Charged Particle
When the charge moves perpendicular to the B-field,
we can show that:
radius rm vqB
period T2 mqB
frequency fqB
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
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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.
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Crossed () Electric and Magnetic Fields
x x x x
x x x x
x
x
B
E
- v
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…
vEB
Web Links: Magnetism inside a TV, TV Screens
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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)
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What happens to a current-carrying wire in a B-field?
Remember: current is just moving charge
B
IL
What is the direction of force on this wire?
We can derive an equation for the
magnitude of this force…
F = I L B sin = angle between B and current
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x x x x
x x x x
x x x x
x x x x
Ex:
x
x
x
x xx
B = .440 T
L
L = 62.0 cm
m = 13.0 g
Find the magnitude and direction of the current that must flow through the red bar in order to
remove the tension from the springs.
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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
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r
Magnetic Field due to a long straight current:
I
BThumb points with IFingers curl with B
Right Hand Rule #2
The magnitude of B depends on the distance r
from the current:
BI
2 r0
0 = 4 x 10-7 Tm/A
permeability of free space
Weblink: Right Hand Rule
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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)
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Parallel Currents
I1 I2
B1 x
x
x
x
d
L
Current I1 produces a B-field
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)
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Consider a circular current…
I
I
I
I
Bx
BB
B
B
B
B
and use RHR #2 to determine the direction of the magnetic field at the center of the loop:
BI
2R0
At the center of the loop:
Radius of loop
or
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I I I
If there are many circular loops:
B NI
2R
0
N = number of loops
Web Link: Compass in loops of current
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Magnetic Fields add as vectors
I
I
I
The straight section creates a B-fieldThe circular section creates a B-field
Do these fields add or
subtract?
I
I
I
At the center of the loop:
Do the B-fields add or subtract in this case?
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Solenoid
x x x x x x x x x x x x
B
inside:
I
I
For a long, ideal solenoid: B = 0n I
n = turns/length
Web Link: Solenoid Factors
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What are solenoids used for?
doorbells
Web Link: How doorbells work
car starters
electric door locks
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Ex:
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?
20 cm
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Toroid
Asteroids
In video games, what does it mean to play in a
“toroidal world”
Web Link: Asteroids
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From above
B
Magnetic Flux () is related to the number of
magnetic field lines passing through a surface
B
NS
B
Web Link: Flux
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Magnetic Flux = = B A cos
B = magnetic fieldA = surface area = angle between B and the
Normal to the surface
SI unit = 1 Weber = T·m2
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Ex: square loop
2.0 mB = 5.0 x 10-4 T
b) Calculate the magnetic flux through the loop
c) What happens to the flux if the normal is rotated by 30° ?
a) What is the angle in this example?
d) What happens to the flux if the normal is rotated by 90° ?
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Recall: An emf is anything that produces a voltage difference (and therefore causes current flow)
I
I
I
I
Bx
B
Recall: For a current loop, we can determine the direction of the B-field at its center:
Here’s a quicker way to do this:
Loop Right Hand Rule Fingers curl with I Thumb points with B
B
I
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Faraday’s Law of Electromagnetic Induction
An emf is induced in a conducting loop whenever the magnetic flux () is changing.
em ft
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
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5) If there are multiple loops:
em f Nt
(N = number of turns)
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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?
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Ex:
NS
B
As the loop moves to the left, what is the direction of the current that is induced in it?
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As loop moves left:
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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
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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
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N S
Ex:
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
B
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Ex:
x x x x
x x x x
x x x x
x x x x
B = 2.0 T
20 cm
20 cm
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.
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Finally…
why does it take so long for a magnet to fall through an aluminum pipe??
Web Link: Lenz’s Law Pipe
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There are many familiar examples of induction
all around us…
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Electric Guitar
Web Link: Electric Guitar
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Motional emf
speed v
conductor
What happens to the positive
charge on the conductor?
What about the negative charge?
Potential difference between the top and
bottom =
x x x x
x x x x
x x x x
x x x x
B
L
Motional emf = vBL
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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?
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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?
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Circuits
DC voltage source
AC voltage source
Resistor
Capacitor
Inductor(Solenoid)
E-field inside
B-field inside
If N = number of turnsI = current = magnetic flux
Inductance = LN
I
SI unit = Henry(H) = Wb/A
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The inductance (L) of a solenoid is not determined by the current or flux through it at a particular moment.
A
Recall:n = turns / length
L = 0 n2 A ℓ
Inductors store energy in their B-fields:
Energy stored in an inductor = ½ L I2
Energy DensityEnergyVolum e
B2
2
0
L is a fixed property of each inductor:
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How do inductors behave in circuits?
BL
+ -
I I Constant B
very boring
Constant I
Changing I Changing B Changing
Induced emf
em f LIt
voltage across inductor
Opposes change in I
Since there is only one inductor, this is called
Self-Induction
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When two inductors affect each other, it is called Mutual-Induction
+ -
1 2
I1
B1
2
N2 turns
If I1 changes
B1 changes
2 changes
emf2 induced in circuit 2
em f MIt21
MN
I2
1
2
Mutual Inductance =
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Primary Circuit
Secondary 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?
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Recall : Power = I V
IV
IV
Current is reduced to minimize power loss
Voltage is reduced to household levels
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How is the power line voltage raised and lowered?
Transformer Station
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Transformer - increases (steps up) or decreases (steps down)
ac voltage using induction
Web Link: Faraday’s Transformer
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Iron
generator
Primary Coil
Voltage VP
NP turns
Secondary Coil
Voltage Vs
NS turns
VV
NN
S
P
S
P
Transformer Equation
Web Link: Transformer
Transformer:
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Ex:
?120 V 3.0 A
Find the output voltage and current.
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Recall the difference between AC and DC:
Web Link: AC vs. DC
DC ( ) :
Voltage
time
Ex:
AC ( ) :
Voltage time
Ex:
V = V0 sin ( 2 f t )
Voltage amplitude frequency time
V0
-V0
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Before we study AC circuits, let’s prepare by reviewing how the circuit components behave in a DC circuit:
I = V/RI
R
V CI
I = V/R at the first instant, then it decreases until I = 0
RV
At this point, the capacitor is fully charged, and acts like a break in the circuit
I
R
V L
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.
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Resistor in an AC Circuit
V = V0sin(2ft) R
II
2rm s0
VV
2rm s0
IV
Rrm srm s
These are all average
valuesWhat about the
instantaneous values?
Web Link: AC Circuits
It
Vt
Voltage and Current are in phase in a purely
resistive circuit.
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Capacitor in an AC Circuit
Acts like a resistor:
R = X1
2 f CC
Capacitive Reactance
SI unit = Ohms ()IVXrm s
rm s
C
What happens to XC when the frequency is very large ??
What happens to XC when the frequency is very small ??
CVrms f
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Instantaneous Values for a Capacitor in an AC Circuit
Web Link: AC Circuits
V
t
I (q/t)
t
Capacitor is full here: q=0 Capacitor is charging
fastest when empty
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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
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L
Inductor in an AC Circuit
Acts like a resistor:
R = X 2 f LL
Inductive Reactance
SI unit = Ohms ()I
VXrm srm s
L
What happens to XL when the frequency is very small ??
What happens to XL when the frequency is very large ??
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Instantaneous Values for an Inductor in an AC Circuit
Web Link: AC CircuitsL
I
t
V (
I/t) t
I is not changing: V=0
I decreasing fastest: V is minimum
I increasing fastest: V is maximum
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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
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Series RCL Circuits
Acts like a resistor:
R = Z R X X2L c
2
IVZrm srm s
Phase Angle between I & V = = tanX X
R1 L C
cos = power factor
Impedance ()
Average Power ( P ) = Irms Vrms cos
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Ex:
16.0
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
15.0 V 1350 Hz
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Non-Series RCL Circuits
Vrms , f
a) Find Irms for a very large frequency
b) Find Irms for a very small frequency
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I
I
Mass on a spring
Resonance in AC Circuits
Oscillating systems:
KE PE PE
AC Circuit
++++
- - - - B-fieldE-fieldWeb Link:
Electromagnetic Oscillating Circuit
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LC This circuit has a natural frequency
f1
2 LC0
Resonant frequency for an RCL circuit(independent of R)
Ex: Tuning a Radio
Web Link: Radio Tuning
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Electromagnetic Wave
Mutually perpendicular and oscillating Electric and Magnetic fields
Web Link: Electromagnetic Wave
Electromagnetic waves travel at the speed of light in a vacuum: c = 3.00 x 108 m/s
Electromagnetic waves are transverse waves
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Recall these facts:
1) A changing B-field produces an E-field
2) A changing E-field produces a B-field-+
atom
E-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
B
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The Electromagnetic (e/m) Spectrum
c = f
speed of light frequency
wavelength Web Link: Wavelengths
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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
0 oDo the calculation.
What do you get?
This is the measured speed of light! Electromagnetic Waves do exist,
and light must be one of them!
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const. velocity
Our Reference Frame determines where and when we observe an event:
x
y
z
x
y
z
In both cases, the Reference Frame is at rest with respect to the observer
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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
in a truck with constant acceleration
Inertial Reference Frames (constant velocity)
Non-Inertial Reference Frame
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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 truck
speed of light
Result
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For speeds far less than c, relativity is barely noticeable
b) Length Contraction (things shrink)
a) Time Dilation (time slows down)
For greater speeds, observers in different reference frames experience:
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Time Dilation
To an observer on the ground, what path
does the light follow?
Now imagine putting it on a spaceship.
Imagine a “light clock”
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tt
1 vc
0
2
2
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 ???
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tt
1 vc
0
2
2
<1
<1
t > t0
Web Link: Time DilationProof:
2) GPS and airplane navigation must use it in their calculations!
1) Atomic clocks on jets slow by precisely this amount
3) Muons arrive at earth’s surface Web Link: Muon Time Dilation
Time slows down in a reference frame that is moving relative to the observer !
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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
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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 ??
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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 ??
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Understanding Time Dilation
x
y
Constant speed in x-direction
More y-motion, less x-motion
time
space
Sitting still (not moving through space)
More motion through space, less motion through time
Just think of time as the 4th dimension
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Length Contraction
L0
Observer (t)
(t0)v
v = relative speed
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
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L L 1 vc0
2
2 Length
Contraction Equation
<1
Web Link: Length Contraction
*Only in the direction of motion:
Distances/lengths appear shorter when moving relative to the observer.
v
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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 ?
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Recall: momentum = p = mv
m1v1 m2
v2
m1v1 + m2v2 = constant
Conservation of Momentum:
When things are moving close to the speed of light, this equation is way off !
We need to consider…
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Relativistic Momentum
pm v
1 vc
2
2
<1
>mv
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?
If we calculate momentum this way for high speeds, conservation of
momentum is obeyed.
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Stanford Linear Particle Accelerator
Electrons accelerate to 99.99999997% speed of light !
Momentum is 40,000 times greater than mv !
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Em c
1 vc
2
2
2
Total Energy of an Object =
E = mc2 Mass-Energy Equivalence
MassEnergy conserved together
If v=0 : E = mc20
= rest energy
This much energy
is equivalent
to
This much mass
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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!
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Our country uses about 3.3 trillion kWhrs of energy per year. Find the amount of mass that is
equivalent to this much energy.
Ex:
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E0 = mc2
If energy changes
Mass must change also
When a 1 kg ball falls 200 m and lands on the ground, by how much does its mass change?
Why don’t we notice this ?
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More examples of Mass-Energy Equivalence…
Ex: Matter meets antimatter
e-
electron
e+
positron
+ =
gamma rays
2 (9.11x10-31 kg) mass = 0
pure energy
People used to wonder if the moon was made of
antimatter
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Ex: Nuclear Power (Fission)
Big nucleus 2 smaller nuclei(less total mass, less energy)
Web Link: Fission
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Ex: The Sun (Fusion)
Two small nuclei
(less total mass, less energy)
Larger nucleus
Web Link: Fusion
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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)
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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 for KE… K E m c
1
1 vc
12
2
2
Relativistic Kinetic Energy
What happens to this equation if an object is traveling at the speed of light?
Objects with mass cannot reach the speed of light
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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?
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Particles experience:
Collisions
Waves experience:
Interference
When they are headed for the same place at the same time…
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Electrons are…
Particles:-
--
and Waves:Interference
Web Links:Electron Interference
Double Slit Experiment
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collisions
Light is…
a Wave:
and a Particle:
light
metal-
-Photoelectric
Effect
Wave-Particle Duality
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Light (any electromagnetic wave) is composed of …
Photons – massless energy particles
E = h f
E = Energy of 1 photon
h = Planck’s constant = 6.626 x 10-34 Js
f = frequency of light wave
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Ex:
How many photons are emitted in 1 hour by a 25 Watt red light bulb ? ( For red, use =750 nm)
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Ex:
Which type of electromagnetic wave is represented by photons with the following energies ?
E = 3.3 x 10-16 J
a)
E = 1.3 x 10-20 J
b)
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The Photoelectric Effect
W0 = Work Function = minimum work
required to eject an electron from the metal
Photon E=hf -
Electron with maximum KE
Web Link: Photoelectric Effect
Conservation of Energy: hf = W0 + KEmax
More light does not result in electrons with more KE
Energy is being absorbed in packets (like particles)
No electrons are ejected if the frequency is too low
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The Photoelectric Effect in the garage…
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More Photoelectric Effect Applications
Photographer’s light meter
Digital Camera
Web Link: Digital Camera
Automatic Doors
Web Link: Solar Energy
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Ex:
Sodium (W0=2.28 eV)
White Light (all colors) = 380-750 nm
-
-
Find the maximum kinetic energy of the ejected electrons (in electron-Volts).
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(Energy=hf)
(Energy=hf’)
The Compton Effect
Web Link: Compton Effect
Does the photon have more or
less energy after the collision?
The electron now has some Kinetic Energy
Photon Mom entum ph
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e
’
e
Conservation of Energy & Conservation of Momentum…
h
m c1 cos m = electron mass
h = Planck’s constant
c = speed of light
hm c
= Compton wavelength = 2.43 x 10-12 m
What is the change in wavelength if =0°? =180°?
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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
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When they finally tried it out with electrons, the interference pattern corresponded perfectly
to this wavelength!
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:
hp
De Broglie Wavelength
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Ex:
Find the de Broglie wavelength of a car with a mass of 1000 kg traveling at a
speed of 30 m/s.
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So what does this wavelength really mean for particles??
It’s a Probability Wave:
100 electrons
3000 electrons70000 electrons
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Does the universe exist if we’re not looking???
Web Link:
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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
x ph
4
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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.
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Ex:
The marble (m=25 g) is somewhere within the box. Find the uncertainty in the marble’s speed.
10 cm
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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.”
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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
E th
4
Web Links: Scanning Tunneling Microscope Animated STM
STM images
This leads to “Quantum Tunneling”
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The best part about knowing all this physics, is that now you
will get the jokes……
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A Party of Famous Physicists
Let’s see how many of the following physicists you can guess…
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Everyone was attracted to his magnetic personality.
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He was under too much pressure to enjoy himself.
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He may or may not have been there.
?
??
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He went back to the buffet table several times a minute.
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He turned out to be a powerful speaker.
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He got a real charge out of the whole thing.
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He thought it was a relatively good time.