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Electric Fields and Forces
AP Physics B
Electric Charge
“Charge” is a property of subatomic particles.
Facts about charge:
 There are 2 types basically, positive (protons)
and negative (electrons)
 LIKE charges REPEL and OPPOSITE
charges ATTRACT
 Charges are symbolic of fluids in that they
can be in 2 states, STATIC or DYNAMIC.
Electric Charge – The specifics
•The symbol for CHARGE is “q”
•The unit is the COULOMB(C),
named after Charles Coulomb
•If we are talking about a SINGLE
charged particle such as 1 electron
or 1 proton we are referring to an
ELEMENTARY charge and often
use, e , to symbolize this.
Some important
constants:
Particle
Charge
Mass
Proton
1.6x10-19 C
1.67 x10-27 kg
Electron
1.6x10-19 C
9.11 x10-31 kg
Neutron
0
1.67 x10-27 kg
Charge is “CONSERVED”
Charge cannot be
created or destroyed
only transferred from
one object to another.
Even though these 2
charges attract initially,
they repel after
touching. Notice the
NET charge stays the
same.
Conductors and Insulators
The movement of charge is limited by the substance
the charge is trying to pass through. There are
generally 2 types of substances.
Conductors: Allow charge to move readily though it.
Insulators: Restrict the movement of the charge
Conductor = Copper Wire
Insulator = Plastic sheath
Charging and Discharging
There are basically 2 ways
you can charge
“BIONIC is the first-ever ionic formula
something.
mascara. The primary ingredient in
1. Charge by friction BIONIC is a chain molecule with a
positive charge. The friction caused by
sweeping the mascara brush across
2. Induction
lashes causes a negative charge. Since
opposites attract, the positively charged
formula adheres to the negatively
charged lashes for a dramatic effect that
lasts all day.”
Induction and Grounding
The second way to charge something is via
INDUCTION, which requires NO PHYSICAL
CONTACT.
We bring a negatively charged rod near a neutral sphere. The protons in the sphere
localize near the rod, while the electrons are repelled to the other side of the sphere. A
wire can then be brought in contact with the negative side and allowed to touch the
GROUND. The electrons will always move towards a more massive objects to increase
separation from other electrons, leaving a NET positive sphere behind.
Electric Force
The electric force between 2 objects is symbolic of the
gravitational force between 2 objects. RECALL:
Fg Mm
1
Fg  2
r
1
q1q2
F

E
r2
r2
k  constant of proportion ality
FE  q1q2
FE 
2
Nm
k  Coulomb constant  8.99 x109 2
C
q1q2
FE  k 2  Coulomb' s Law
r
Electric Forces and Newton’s Laws
Electric Forces and Fields obey Newton’s Laws.
Example: An electron is released above the
surface of the Earth. A second electron
directly below it exerts an electrostatic
force on the first electron just great enough
to cancel out the gravitational force on it.
How far below the first electron is the
second?
FE  mg
q1q2
q1q2
k 2  mg  r  k
r
mg
Fe
e
mg
e
r=?
19 2
(
1
.
6
x
10
)
9
(8.99 )

31
(9.11x10 )(9.8)
5.1 m
Electric Forces and Vectors
Electric Fields and Forces are ALL vectors,
thus all rules applying to vectors must be
followed.
Consider three point charges, q1 = 6.00 x10-9 C (located at the origin),q3 =
5.00x10-9 C, and q2 = -2.00x10-9 C, located at the corners of a RIGHT triangle.
q2 is located at y= 3 m while q3 is located 4m to the right of q2. Find the
resultant force on q3.
Which way does q2 push q3?
4m
q2
q3
Which way does q1 push q3?
3m
q1
q
Fon 3 due to 1
5m
Fon 3 due to 2
q3
q= tan-1(3/4)
q = 37
Example Cont’
4m
q2
3m
q1
q
q3
Fon 3 due to 1
5m
Fon 3 due to 2
q= tan-1(3/4)
q = 37
q3
F3,1sin37
F3,1cos37
F3, 2
F3, 2
(5.0 x10 9 )( 2 x10 9 )
 (8.99 x10 )
42
 5.6 x10-9 N
9
(6 x10 9 )(5 x10 9 )
F3,1  (8.99 x10 )
52
F3,1  1.1x10-8 N
9
F
F
F
x
 F3,1 cos(37)  F3, 2
x
 3.18 x10 9 N
y
 F3,1 sin( 37)  6.62 x10 9 N
Fresultant  ( Fx ) 2  ( Fy ) 2
Fres  7.34x10-9 N
Direction  q  tan
1
F

(
F
y
)
x
64.3 degrees above the +x
Electric Fields
By definition, the are
“LINES OF FORCE”
Some important facts:
 An electric field is a
vector
 Always is in the direction
that a POSITIVE “test”
charge would move
 The amount of force
PER “test” charge
If you placed a 2nd positive charge
(test charge), near the positive
charge shown above, it would
move AWAY.
If you placed that same charge
near the negative charge shown
above it would move TOWARDS.
Electric Fields and Newton’s Laws
Once again, the equation for
ELECTRIC FIELD is
symbolic of the equation for
WEIGHT just like
coulomb’s law is symbolic
of Newton’s Law of
Gravitation.
The symbol for Electric Field is, “E”. And since it is defined as a force per
unit charge he unit is Newtons per Coulomb, N/C.
NOTE: the equations above will ONLY help you determine the MAGNITUDE
of the field or force. Conceptual understanding will help you determine the
direction.
The “q” in the equation is that of a “test charge”.
Example
An electron and proton are each placed at rest in an external
field of 520 N/C. Calculate the speed of each particle after


48 ns
 FE
FE
E
520 
What do we know
q
1.6 x1019
-19 N
F

8.32
x10
E
-31
me=9.11 x 10
kg
mp= 1.67 x10-27 kg
qboth=1.6
x10-19
C
FNet  ma FE  FNet
FE  me a  (9.11x10 31 )a 
vo = 0 m/s
FE  m p a  (1.67 x10  27 )a 
E = 520 N/C
v  vo  at
t = 48 x 10-9 s
ve  ae (48 x10 9 ) 
4.38 x106 m/s
v p  a p ( 48 x10 9 ) 
2.39 x103 m/s
9.13x1013 m/s/s
4.98 x1010 m/s/s
An Electric Point Charge
As we have discussed, all charges exert forces on other charges
due to a field around them. Suppose we want to know how
strong the field is at a specific point in space near this charge
the calculate the effects this charge will have on other charges
should they be placed at that point.
Qq
FE  k 2
r
Qq
Eq  k 2
r
Epoint charge
POINT CHARGE
FE
E
 FE  Eq
q
kQ
 2
r
TEST CHARGE
Example
A -4x10-12C charge Q is placed at the origin. What is the
magnitude and direction of the electric field produced
by Q if a test charge were placed at x = -0.2 m ?
kQ
9 ( 4 x10
E  2  8.99 x10
2
r
.2
Emag  0.899 N/C
Edir 
Towards Q to the right
12
)
0.2 m
E
E
-Q
E
E
Remember, our equations will only give us MAGNITUDE. And the electric
field LEAVES POSITIVE and ENTERS NEGATIVE.
Electric Field of a Conductor
A few more things about electric fields, suppose you bring a conductor
NEAR a charged object. The side closest to which ever charge will be
INDUCED the opposite charge. However, the charge will ONLY exist
on the surface. There will never be an electric field inside a conductor.
Insulators, however, can store the charge inside.
There must be a
positive charge on
this side
There must be a
negative charge on
this side OR this
side was induced
positive due to the
other side being
negative.
Electric Circuits
AP Physics B
Potential Difference =Voltage=EMF
In a battery, a series of chemical
reactions occur in which
electrons are transferred from
one terminal to another. There is
a potential difference (voltage)
between these poles.
The maximum potential difference
a power source can have is
called the electromotive force
or (EMF), e. The term isn't
actually a force, simply the
amount of energy per charge
(J/C or V)
A Basic Circuit
All electric circuits have three main parts
1.
2.
3.
A source of energy
A closed path
A device which uses the energy
If ANY part of the circuit is open the device will not work!
Electricity can be symbolic of Fluids
Circuits are very similar to water flowing through a pipe
A pump basically works on TWO
IMPORTANT PRINCIPLES concerning its
flow
•
•
There is a PRESSURE DIFFERENCE
where the flow begins and ends
A certain AMOUNT of flow passes each
SECOND.
A circuit basically works on TWO
IMPORTANT PRINCIPLES
•
•
There is a "POTENTIAL DIFFERENCE
aka VOLTAGE" from where the charge
begins to where it ends
The AMOUNT of CHARGE that flows
PER SECOND is called CURRENT.
Current
Current is defined as the rate at which charge
flows through a surface.
The current is in the same direction as the flow
of positive charge (for this course)
Note: The “I” stands
for intensity
There are 2 types of Current
DC = Direct Current - current flows in one direction
Example: Battery
AC = Alternating Current- current reverses direction many times per second.
This suggests that AC devices turn OFF and
ON. Example: Wall outlet (progress energy)
Ohm’s Law
“The voltage (potential difference, emf) is directly
related to the current, when the resistance is
constant”
10
9
8
7
Voltage(V)
V I
R  constant of proportion ality
R  Resistance
V  IR
e  IR
Voltage vs. Current
6
5
Voltage(V)
4
3
2
1
Since R=V/I, the resistance is the
SLOPE of a V vs. I graph
0
0
0.2
0.4
0.6
Current(Amps)
0.8
1
Resistance
Resistance (R) – is defined as the restriction of electron
flow. It is due to interactions that occur at the atomic
scale. For example, as electron move through a
conductor they are attracted to the protons on the
nucleus of the conductor itself. This attraction doesn’t
stop the electrons, just slow them down a bit and cause
the system to waste energy.
The unit for resistance is
the OHM, W
Electrical POWER
We have already learned that POWER is the rate at which work
(energy) is done. Circuits that are a prime example of this as
batteries only last for a certain amount of time AND we get
charged an energy bill each month based on the amount of
energy we used over the course of a month…aka POWER.
POWER
It is interesting to see how certain electrical
variables can be used to get POWER. Let’s
take Voltage and Current for example.
Other useful power formulas
These formulas can also
be used! They are
simply derivations of
the POWER formula
with different versions
of Ohm's law
substituted in.
Ways to Wire Circuits
There are 2 basic ways to wire a circuit. Keep in
mind that a resistor could be ANYTHING ( bulb,
toaster, ceramic material…etc)
Series – One after another
Parallel – between a set of junctions and
parallel to each other
Schematic Symbols
Before you begin to understand circuits you need to be able to
draw what they look like using a set of standard symbols
understood anywhere in the world
For the battery symbol, the
LONG line is considered to be
the POSITIVE terminal and the
SHORT line , NEGATIVE.
The VOLTMETER and AMMETER
are special devices you place IN
or AROUND the circuit to
measure the VOLTAGE and
CURRENT.
The Voltmeter and Ammeter
Current goes THROUGH the ammeter
The voltmeter and ammeter cannot be
just placed anywhere in the circuit. They
must be used according to their
DEFINITION.
Since a voltmeter measures voltage or
POTENTIAL DIFFERENCE it must be
placed ACROSS the device you want
to measure. That way you can measure
the CHANGE on either side of the
device.
Voltmeter is drawn ACROSS the resistor
Since the ammeter measures the current or
FLOW it must be placed in such a way as the
charges go THROUGH the device.
Simple Circuit
When you are drawing a
circuit it may be a wise
thing to start by drawing
the battery first, then
follow along the loop
(closed) starting with
positive and drawing
what you see.
Series Circuit
In in series circuit, the resistors
are wired one after another.
Since they are all part of the
SAME LOOP they each
experience the SAME
AMOUNT of current. In
figure, however, you see
that they all exist
BETWEEN the terminals of
the battery, meaning they
SHARE the potential
(voltage).
I ( series)Total  I1  I 2  I 3
V( series)Total  V1  V2  V3
Series Circuit
I ( series)Total  I1  I 2  I 3
V( series)Total  V1  V2  V3
As the current goes through the circuit, the charges must USE ENERGY to get
through the resistor. So each individual resistor will get its own individual potential
voltage). We call this VOLTAGE DROP.
V( series)Total  V1  V2  V3 ; V  IR
( I T RT ) series  I1 R1  I 2 R2  I 3 R3
Rseries  R1  R2  R3
Rs   Ri
Note: They may use the
terms “effective” or
“equivalent” to mean
TOTAL!
Example
A series circuit is shown to the left.
a) What is the total resistance?
R(series) = 1 + 2 + 3 = 6W
b)
What is the total current?
V=IR
V1W(2)(1) 2 V
12=I(6)
I = 2A
c)
What is the current across EACH
resistor? They EACH get 2 amps!
d)
What is the voltage drop across
each resistor?( Apply Ohm's law
to each resistor separately)
V3W=(2)(3)= 6V
V2W=(2)(2)= 4V
Notice that the individual VOLTAGE DROPS add up to the TOTAL!!
Parallel Circuit
In a parallel circuit, we have
multiple loops. So the
current splits up among
the loops with the
individual loop currents
adding to the total
current
It is important to understand that parallel
circuits will all have some position
where the current splits and comes back
together. We call these JUNCTIONS.
The current going IN to a junction will
always equal the current going OUT of a
junction.
I ( parallel)Total  I1  I 2  I 3
Junctions
Regarding Junctions :
I IN  I OUT
Parallel Circuit
V
Notice that the JUNCTIONS both touch the
POSTIVE and NEGATIVE terminals of the
battery. That means you have the SAME
potential difference down EACH individual
branch of the parallel circuit. This means
that the individual voltages drops are equal.
V( parallel)Total  V1  V2  V3
I ( parallel)Total  I1  I 2  I 3 ; V  IR
This junction
touches the
POSITIVE
terminal
This junction
touches the
NEGATIVE
terminal
VT
V1 V2 V3
( ) Parallel  

RT
R1 R2 R3
1
1
1
1
 

RP R1 R2 R3
1
1

RP
Ri
Example
To the left is an example of a parallel circuit.
a) What is the total resistance?
1 1 1 1
  
RP 5 7 9
2.20 W
1
1
 0.454  RP 

Rp
0.454
b) What is the total current? V  IR
8  I ( R )  3.64 A
c) What is the voltage across EACH resistor?
8 V each!
d) What is the current drop across each resistor?
(Apply Ohm's law to each resistor separately)
V  IR
8
8
8
I 5W   1.6 A I 7 W  1.14 A I 9W   0.90 A
5
7
9
Notice that the
individual currents
ADD to the total.
Compound (Complex) Circuits
Many times you will have series and parallel in the SAME circuit.
Solve this type of circuit
from the inside out.
WHAT IS THE TOTAL
RESISTANCE?
1
1
1

 ; RP  33.3W
RP 100 50
Rs  80  33.3  113.3W
Compound (Complex) Circuits
1
1
1

 ; RP  33.3W
RP 100 50
Rs  80  33.3  113.3W
Suppose the potential difference (voltage) is equal to 120V. What is the total
current?
VT  I T RT
120  I T (113.3)
I T  1.06 A
What is the VOLTAGE DROP across the 80W resistor?
V80W  I 80W R80W
V80W  (1.06)(80)
V80W 
84.8 V
Compound (Complex) Circuits
RT  113.3W
VT  120V
I T  1.06 A
V80W  84.8V
I 80W  1.06 A
What is the VOLTAGE DROP across
the 100W and 50W resistor?
VT ( parallel)  V2  V3
VT ( series)  V1  V2&3
120  84.8  V2&3
V2&3  35.2 V Each!
What is the current across the
100W and 50W resistor?
I T ( parallel)  I 2  I 3
I T ( series)  I1  I 2&3
35.2 0.352 A
I100W 

100
35.2
I 50W 
 0.704 A
50
Add to
1.06A
Electrical Energy, Potential
and Capacitance
AP Physics B
Electric Fields and WORK
In order to bring two like charges near each other work must be
done. In order to separate two opposite charges, work must be
done. Remember that whenever work gets done, energy
changes form.
As the monkey does work on the positive charge, he increases the energy of
that charge. The closer he brings it, the more electrical potential energy it
has. When he releases the charge, work gets done on the charge which
changes its energy from electrical potential energy to kinetic energy. Every
time he brings the charge back, he does work on the charge. If he brought
the charge closer to the other object, it would have more electrical potential
energy. If he brought 2 or 3 charges instead of one, then he would have had
to do more work so he would have created more electrical potential
energy. Electrical potential energy could be measured in Joules just like any
other form of energy.
Electric Fields and WORK
Consider a negative charge moving
in between 2 oppositely charged
parallel plates initial KE=0 Final
KE= 0, therefore in this case
Work = PE
We call this ELECTRICAL potential
energy, UE, and it is equal to the
amount of work done by the
ELECTRIC FORCE, caused by the
ELECTRIC FIELD over distance, d,
which in this case is the plate
separation distance.
Is there a symbolic relationship with the FORMULA for gravitational
potential energy?
Electric Potential
U g  mgh
Here we see the equation for gravitational
potential energy.
U g  U E (or W )
Instead of gravitational potential energy we are
talking about ELECTRIC POTENTIAL ENERGY
mq
A charge will be in the field instead of a mass
gE
hxd
U E (W )  qEd
W
 Ed
q
The field will be an ELECTRIC FIELD instead of
a gravitational field
The displacement is the same in any reference
frame and use various symbols
Putting it all together!
Question: What does the LEFT side of the equation
mean in words? The amount of Energy per charge!
Energy per charge
The amount of energy per charge has a specific
name and it is called, VOLTAGE or ELECTRIC
POTENTIAL (difference). Why the “difference”?
1 mv2
W K
V  
 2
q
q
q
Understanding “Difference”
Let’s say we have a proton placed
between a set of charged plates. If
the proton is held fixed at the
positive plate, the ELECTRIC
FIELD will apply a FORCE on the
proton (charge). Since like charges
repel, the proton is considered to
have a high potential (voltage)
similar to being above the ground.
It moves towards the negative plate
or low potential (voltage). The
plates are charged using a battery
source where one side is positive
and the other is negative. The
positive side is at 9V, for example,
and the negative side is at 0V. So
basically the charge travels through
a “change in voltage” much like a
falling mass experiences a
“change in height. (Note: The
electron does the opposite)
BEWARE!!!!!!
W is Electric Potential Energy (Joules)
is not
V is Electric Potential (Joules/Coulomb)
a.k.a Voltage, Potential Difference
The “other side” of that equation?
U g  mgh
U g  U E (or W )
mq
gE
hxd
U E (W )  qEd
W
 Ed
q
Since the amount of energy per charge is
called Electric Potential, or Voltage, the
product of the electric field and
displacement is also VOLTAGE
This makes sense as it is applied usually
to a set of PARALLEL PLATES.
V=Ed
V
E
d
Example
A pair of oppositely charged, parallel plates are separated by
5.33 mm. A potential difference of 600 V exists between the
plates. (a) What is the magnitude of the electric field strength
between the plates? (b) What is the magnitude of the force
on an electron between the plates?
d  0.00533m
V  600V
E ?
qe   1.6 x10 19 C
V  Ed
600  E (0.0053)
E  113,207.55 N/C
Fe
Fe
E

q 1.6 x10 19 C
Fe  1.81x10-14 N
Example
Calculate the speed of a proton that is accelerated
from rest through a potential difference of 120 V
q p   1.6 x10 19 C
m p   1.67 x10  27 kg
V  120V
v?
W K
V 


q
q
1
2mv2
q
2qV
2(1.6 x10 19 )(120) 1.52x105 m/s
v


 27
m
1.67 x10
Electric Potential of a Point Charge
Up to this point we have focused our attention solely to
that of a set of parallel plates. But those are not the
ONLY thing that has an electric field. Remember,
point charges have an electric field that surrounds
them.
So imagine placing a TEST
CHARGE out way from the
point charge. Will it experience
a change in electric potential
energy? YES!
Thus is also must experience a
change in electric potential as
well.
Electric Potential
Let’s use our “plate” analogy. Suppose we had a set of parallel plates
symbolic of being “above the ground” which has potential difference of
50V and a CONSTANT Electric Field.
+++++++++++
V = ? From 1 to 2
1
25 V
V = ? From 2 to 3
d
E
2
3
0V
0.5d, V= 25 V
V = ? From 3 to 4
12.5 V
4
0.25d, V= 12.5 V
V = ? From 1 to 4
37.5 V
---------------Notice that the “ELECTRIC POTENTIAL” (Voltage) DOES NOT change from
2 to 3. They are symbolically at the same height and thus at the same voltage.
The line they are on is called an EQUIPOTENTIAL LINE. What do you notice
about the orientation between the electric field lines and the equipotential
lines?
Equipotential Lines
So let’s say you had a positive
charge. The electric field lines
move AWAY from the charge.
The equipotential lines are
perpendicular to the electric
field lines and thus make
concentric circles around the
charge. As you move AWAY
from a positive charge the
potential decreases. So
V1>V2>V3.
Now that we have the direction or
visual aspect of the
equipotential line understood
the question is how can we
determine the potential at a
certain distance away from the
charge?
r
V(r) = ?
Electric Potential of a Point Charge
Why the “sum” sign?
W FE x
V 

; xr
q
q
FE r
Qq
V 
; FE  k 2
q
r
Qqr
Q
V  k 2  k 
qr
r
Voltage, unlike Electric Field,
is NOT a vector! So if you
have MORE than one
charge you don’t need to
use vectors. Simply add up
all the voltages that each
charge contributes since
voltage is a SCALAR.
WARNING! You must use
the “sign” of the charge in
this case.
Potential of a point charge
Suppose we had 4 charges
each at the corners of a
square with sides equal to d.
d 2
If I wanted to find the potential
at the CENTER I would SUM
up all of the individual
potentials.
Thus the distance from
a corner to the center
will equal :
d 2
r
2
Q
r
q 2q  3q 5q
 k( 

 )
r r
r
r
5q
10q
5 2
 k( )  k(
)k
r
d
d 2
Vcenter  k 
Vcenter
Vcenter
Electric field at the center? ( Not so easy)
If they had asked us to find the
electric field, we first would
have to figure out the visual
direction, use vectors to break
individual electric fields into
components and use the
Pythagorean Theorem to find
the resultant and inverse
tangent to find the angle
Eresultant
So, yea….Electric Potentials are
NICE to deal with!
Example
An electric dipole consists of two charges q1 = +12nC and q2
= -12nC, placed 10 cm apart as shown in the figure.
Compute the potential at points a,b, and c.
q1 q2
Va  k  (  )
ra ra
12 x109  12 x109
Va  8.99 x10 (

)
0.06
0.04
Va  -899 V
9
Example cont’
Vb  k  (
q1 q2
 )
rb rb
9
9
12
x
10

12
x
10
Vb  8.99 x109 (

)
0.04
0.14
Vb  1926.4 V
Vc 
0V
Since direction isn’t important, the
electric potential at “c” is zero. The
electric field however is NOT. The
electric field would point to the right.
Applications of Electric Potential
Is there any way we can use a set of plates with an electric
field? YES! We can make what is called a Parallel Plate
Capacitor and Store Charges between the plates!
Storing Charges- Capacitors
A capacitor consists of 2 conductors
of any shape placed near one another
without touching. It is common; to fill
up the region between these 2
conductors with an insulating material
called a dielectric. We charge these
plates with opposing charges to
set up an electric field.
Capacitors in Kodak Cameras
Capacitors can be easily purchased at a
local Radio Shack and are commonly
found in disposable Kodak Cameras.
When a voltage is applied to an empty
capacitor, current flows through the
capacitor and each side of the capacitor
becomes charged. The two sides have
equal and opposite charges. When the
capacitor is fully charged, the current
stops flowing. The collected charge is
then ready to be discharged and when
you press the flash it discharges very
quickly released it in the form of light.
Cylindrical Capacitor
Capacitance
In the picture below, the capacitor is symbolized by a set of parallel
lines. Once it's charged, the capacitor has the same voltage as
the battery (1.5 volts on the battery means 1.5 volts on the
capacitor) The difference between a capacitor and a battery is
that a capacitor can dump its entire charge in a tiny fraction of a
second, where a battery would take minutes to completely
discharge itself. That's why the electronic flash on a camera uses
a capacitor -- the battery charges up the flash's capacitor over
several seconds, and then the capacitor dumps the full charge
into the flash tube almost instantly
Measuring Capacitance
Let’s go back to thinking about plates!
V  Ed ,
The unit for capacitance is the FARAD, F.
V E , if d  constant
E Q Therefore
Q V
C  contant of proportion ality
C  Capacitanc e
Q  CV
Q
C
V
Capacitor Geometry
The capacitance of a
capacitor depends on
HOW you make it.
1
C A C
d
A  area of plate
d  distance beteween plates
A
C
d
e o  constant of proportion ality
e o  vacuum permittivi ty constant
e o  8.85 x10
C
eo A
d
12
C2
Nm 2
Capacitor Problems
What is the AREA of a 1F capacitor that has a plate
separation of 1 mm?
A
C  eo
D
1  8.85 x10
A
Is this a practical capacitor to build?
NO! – How can you build this then?
12
A
0.001
1.13x108 m2
Sides 
10629 m
The answer lies in REDUCING the
AREA. But you must have a
CAPACITANCE of 1 F. How can
you keep the capacitance at 1 F
and reduce the Area at the same
time?
Add a DIELECTRIC!!!
Dielectric
Remember, the dielectric is an insulating material placed
between the conductors to help store the charge. In the
previous example we assumed there was NO dielectric and
thus a vacuum between the plates.
A
C  ke o
d
k  Dielectric
All insulating materials have a dielectric
constant associated with it. Here now
you can reduce the AREA and use a
LARGE dielectric to establish the
capacitance at 1 F.
Using MORE than 1 capacitor
Let’s say you decide that
1 capacitor will not be
enough to build what
you need to build. You
may need to use more
than 1. There are 2
basic ways to assemble
them together
 Series – One after
another
 Parallel – between a set
of junctions and parallel
to each other.
Capacitors in Series
Capacitors in series each charge each other by INDUCTION. So
they each have the SAME charge. The electric potential on the
other hand is divided up amongst them. In other words, the sum
of the individual voltages will equal the total voltage of the battery
or power source.
Capacitors in Parallel
In a parallel configuration, the voltage is the same
because ALL THREE capacitors touch BOTH ends
of the battery. As a result, they split up the charge
amongst them.
Capacitors “STORE” energy
Anytime you have a situation where energy is “STORED” it is
called POTENTIAL. In this case we have capacitor potential
energy, Uc
Suppose we plot a V vs. Q graph.
If we wanted to find the AREA we
would MULTIPLY the 2 variables
according to the equation for Area.
A = bh
When we do this we get Area =
VQ
Let’s do a unit check!
Voltage = Joules/Coulomb
Charge = Coulombs
Area = ENERGY
Potential Energy of a Capacitor
Since the AREA under the line is a
triangle, the ENERGY(area) =1/2VQ
Q
1
U C  VQ C 
2
V
This energy or area is referred
as the potential energy stored
inside a capacitor.
U C  1 V (VC )  1 CV 2
2
2
2
Q
Q
U C  1 ( )Q 
2 C
2C
Note: The slope of the line is
the inverse of the capacitance.
most common form
ElectroMagnetic Induction
AP Physics B
What is E/M Induction?
Electromagnetic Induction is the
process of using magnetic fields to
produce voltage, and in a
complete circuit, a current.
Michael Faraday first discovered it, using some of the works of Hans
Christian Oersted. His work started at first using different combinations of
wires and magnetic strengths and currents, but it wasn't until he tried moving
the wires that he got any success.
It turns out that electromagnetic induction is created by just that - the moving
of a conductive substance through a magnetic field.
Magnetic Induction
As the magnet moves back and forth a current is said
to be INDUCED in the wire.
Magnetic Flux
The first step to understanding the complex nature of
electromagnetic induction is to understand the idea
B
of magnetic flux.
A
Flux is a general term
associated with a FIELD that is
bound by a certain AREA. So
MAGNETIC FLUX is any AREA
that has a MAGNETIC FIELD
passing through it.
We generally define an AREA vector as one that is perpendicular to the
surface of the material. Therefore, you can see in the figure that the
AREA vector and the Magnetic Field vector are PARALLEL. This then
produces a DOT PRODUCT between the 2 variables that then define
flux.
Magnetic Flux – The DOT product
B  B  A
 B  BA cos q
Unit : Tm 2 or Weber(Wb)
How could we CHANGE the flux over a period of time?
 We could move the magnet away or towards (or the wire)
 We could increase or decrease the area
 We could ROTATE the wire along an axis that is PERPENDICULAR to the
field thus changing the angle between the area and magnetic field vectors.
Faraday’s Law
Faraday learned that if you change any part of the flux over time
you could induce a current in a conductor and thus create a
source of EMF (voltage, potential difference). Since we are
dealing with time here were a talking about the RATE of
CHANGE of FLUX, which is called Faraday’s Law.
 B
( BA cos q )
e  N
 N
t
t
N # turns of wire
Useful Applications
The Forever Flashlight uses the Faraday Principle of
Electromagnetic Energy to eliminate the need for batteries. The
Faraday Principle states that if an electric conductor, like copper
wire, is moved through a magnetic field, electric current will be
generated and flow into the conductor.
Useful Applications
AC Generators use
Faraday’s law to produce
rotation and thus convert
electrical and magnetic
energy into rotational
kinetic energy. This idea
can be used to run all kinds
of motors. Since the
current in the coil is AC, it
is turning on and off thus
creating a CHANGING
magnetic field of its own.
Its own magnetic field
interferes with the shown
magnetic field to produce
rotation.
Transformers
Probably one of the greatest inventions of all time is the
transformer. AC Current from the primary coil moves quickly
BACK and FORTH (thus the idea of changing!) across the
secondary coil. The moving magnetic field caused by the
changing field (flux) induces a current in the secondary coil.
If the secondary coil has MORE turns
than the primary you can step up the
voltage and runs devices that would
normally need MORE voltage than
what you have coming in. We call this
a STEP UP transformer.
We can use this idea in reverse as well
to create a STEP DOWN transformer.
Microphones
A microphone works when sound
waves enter the filter of a
microphone. Inside the filter, a
diaphragm is vibrated by the
sound waves which in turn moves
a coil of wire wrapped around a
magnet. The movement of the wire
in the magnetic field induces a
current in the wire. Thus sound
waves can be turned into
electronic signals and then
amplified through a speaker.
Example
A coil with 200 turns of wire is wrapped on an 18.0 cm square frame.
Each turn has the same area, equal to that of the frame, and the
total resistance of the coil is 2.0W . A uniform magnetic field is
applied perpendicularly to the plane of the coil. If the field changes
uniformly from 0 to 0.500 T in 0.80 s, find the magnitude of the
induced emf in the coil while the field has changed as well as the
magnitude of the induced current.
 B
BA cos q
N
t
t
(0.500  0)(0.18 x0.18) cos 90
e  200
0.80
e  4.05 V
e N
e  IR  I (2)
I  2.03 A
Why did you find the
ABSOLUTE VALUE of the
EMF?
What happened to the “ –
“ that was there originally?
Lenz’s Law
Lenz's law gives the direction of the induced emf and current
resulting from electromagnetic induction. The law provides a
physical interpretation of the choice of sign in Faraday's law of
induction, indicating that the induced emf and the change in flux
have opposite signs.
 B
e


N
Lenz’s Law
t
In the figure above, we see that the direction of the current changes. Lenz’s
Law helps us determine the DIRECTION of that current.
Lenz’s Law & Faraday’s Lawe   N 
B
t
Let’s consider a magnet with it’s north pole
moving TOWARDS a conducting loop.
DOES THE FLUX CHANGE? Yes!
DOES THE FLUX INCREASE OR DECREASE?
Increase
WHAT SIGN DOES THE “” GIVE YOU IN
FARADAY’S LAW?
Positive
Binduced
DOES LENZ’S LAW CANCEL OUT? NO
What does this mean?
This means that the INDUCED MAGNETIC FIELD around the WIRE caused
by the moving magnet OPPOSES the original magnetic field. Since the
original B field is downward, the induced field is upward! We then use the
curling right hand rule to determine the direction of the current.
Lenz’s Law
A magnet is
dropped down a
conducting tube.
The INDUCED current creates an INDUCED
magnetic field of its own inside the conductor
that opposes the original magnetic field.
The magnet INDUCES a
current above and below the
magnet as it moves.
Since the induced
field opposes the
direction of the
original it attracts
the magnet upward
slowing the motion
caused by gravity
downward.
If the motion of the magnet were NOT slowed this would violate conservation of energy!
Lenz’s Law
 B
e  N
t
Let’s consider a magnet with it’s north pole
moving AWAY from a conducting loop.
DOES THE FLUX CHANGE? Yes!
DOES THE FLUX INCREASE OR DECREASE?
Decreases
WHAT SIGN DOES THE “” GIVE YOU IN
FARADAY’S LAW?
negative
DOES LENZ’S LAW CANCEL OUT? yes
Binduced
What does this mean?
In this case, the induced field DOES NOT oppose the original and points in
the same direction. Once again use your curled right hand rule to determine
the DIRECTION of the current.
In summary
Faraday’s Law is basically used to find the
MAGNITUDE of the induced EMF. The
magnitude of the current can then be found
using Ohm’s Law provided we know the
conductor’s resistance.
Lenz’s Law is part of Faraday’s Law and can
help you determine the direction of the
current provided you know HOW the flux is
changing
Motional EMF – The Rail Gun
A railgun consists of two parallel metal rails (hence the name) connected to an
electrical power supply. When a conductive projectile is inserted between the rails
(from the end connected to the power supply), it completes the circuit. Electrons
flow from the negative terminal of the power supply up the negative rail, across the
projectile, and down the positive rail, back to the power supply.
In accordance with the right-hand rule,
the magnetic field circulates around
each conductor. Since the current is in
opposite direction along each rail, the
net magnetic field between the rails (B)
is directed vertically. In combination with
the current (I) across the projectile, this
produces a magnetic force which
accelerates the projectile along the rails.
There are also forces acting on the rails
attempting to push them apart, but since
the rails are firmly mounted, they cannot
move. The projectile slides up the rails
away from the end with the power
supply.
Motional Emf
There are many situations where motional EMF can occur that are
different from the rail gun. Suppose a bar of length, L, is pulled to
right at a speed, v, in a magnetic field, B, directed into the page. The
conducting rod itself completes a circuit across a set of parallel
conducting rails with a resistor mounted between them.
e
e
e
I
 B
 N
t
BA
Blx


; e  Blv
t
t
 IR
Blv

R
Motional EMF
In the figure, we are
applying a force this time
to the rod. Due to Lenz’s
Law the magnetic force
opposes the applied
force. Since we know
that the magnetic force
acts to the left and the
magnetic field acts into
the page, we can use the
RHR to determine the
direction of the current
around the loop and the
resistor.
Example
An airplane with a wing span of 30.0 m flies parallel to the Earth’s
surface at a location where the downward component of the
Earth’s magnetic field is 0.60 x10-4 T. Find the difference in
potential between the wing tips is the speed of the plane is 250
m/s.
e  Blv
e  0.60 x104 (30)( 250)
e  0.45 V
In 1996, NASA conducted an experiment with a 20,000-meter conducting
tether. When the tether was fully deployed during this test, the orbiting
tether generated a potential of 3,500 volts. This conducting single-line
tether was severed after five hours of deployment. It is believed that the
failure was caused by an electric arc generated by the conductive tether's
movement through the Earth's magnetic field.