Download Electric Potential

Electric Potential
Unit 2
Energy in Physics
• Energy can be used to study a wider
range of phenomena and problems that
cannot be easily analyzed using forces.
• Energy is also nice because it is a
scalar rather than a vector.
• In this unit, we will look at how energy is
applied in electrostatics.
Electric Potential Energy
• Recall from last year that the change in
an object’s potential energy between
two points is equal to the negative work
done by the conservative force acting
on the object.
U  Ub Ua  Wba
Example: Proton in an Electric Field
A proton is placed in a uniform electric field of 20000 N/C.
It experiences a force to the right.
a) What is the magnitude of the force being exerted
on the proton by the field?
b) How much work does the field do on the proton
if it moves the proton 3.0 cm?
c) What was the change in the proton’s potential
energy?
This is actually a general result for a particle in a uniform
electric field.
U  qEd
Electric Potential Energy
• Notice that, just like with gravitational
potential energy, we are free to set
U = 0 wherever we please.
• Therefore, it is only meaningful to talk
about differences in potential energy.
• This we will also be true for electric
potential.
Electric Potential
Electric Potential
• In the last unit, we ran into a problem
with the “action-at-a-distance” idea of a
force.
• This problem would translate to energy,
since it is related to force.
• To deal with this, we introduced the
electric field, which is the force per unit
charge.
Electric Potential
• Similarly, we will introduce electric
potential to help deal with energy.
• Electric potential is the electric potential
energy per unit charge.
Electric
potential at
point a
Ua
Va 
q
Electric PE
at point a
Charge of
the object
Electric Potential
• Like potential energy, electric potential
must be defined relative to a zero.
• That means that only differences in
potential have any real meaning.
Ub  Ua
Vba  Vb  Va 
q
Electric Potential
• We prefer to use electric potential
instead of potential energy because V
(like E) does not depend on the size of
the test charge.
• V is created by the charges creating E,
and q gains potential energy by being
placed in the potential.
Properties of Potential
• Positive charges move from high V to
low V.
• Negative charges do the reverse.
• Potential is has units of volts (V).
J
1V  1
C
• The word voltage is used to mean
potential difference.

Problem Solving
• Just as we often found the electric force
from the electric field, we will generally
use voltage to find electric potential
energy.
• This can be done by rearranging our
definition of potential.
U  qVb  Va   qVba
Zero Volts
• We can define V = 0 at any place.
However, there are conventions for
where to chose.
• One common choice is to set 0V at the
ground. Any object that is grounded is
also at 0V.
• The other choice is to set V = 0 at
infinity. More on this next week.
The Electron Volt
• We noticed yesterday that the amount
of energy involved in the movement of
electrons and protons is very small.
• The joule is too large a unit to effectively
describe the energy of electrons.
• Notice that we get units of energy when
we multiply a charge times a potential
difference.
The Electron Volt
• We can use this fact to define a new
unit of energy.
• An electron volt (eV) is the amount of
energy an electron receives when
moving through a potential of 1 volt.
1eV  1.6  10
19
J
Example: Moving a Proton
• How much work is done by the electric
field moving a proton from a 100 V to 20 V?
Electron in a TV Tube
The picture tube of a television set has a potential difference
of 5000 V. An electron is accelerated from rest through this
potential difference.
a) What is the change in potential energy of the
electron?
b) What is the speed of the electron when it reaches
the end of the tube?
5000 V
-
+
Homework
• Do problems 1-4, 8, and 11 on page
489.
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Potential vs. Potential Energy
An Analogy
• Suppose we decided to drop a rock off
a cliff.
• The higher the cliff, the more potential
energy the rock would have (U = mgh).
• Remember, potential energy is a
measure of how much KE the object
would gain as a result of the work done
by gravity.
An Analogy
• The actual amount of KE the object
gains depends not only on height, but
also on the mass of the object.
• We could drop a large rock and a small
rock from the same height, but the large
rock would have more PE.
• The height would be considered to be
the gravitational potential.
An Analogy
• The electrical case is similar. Two
charged particles can be at the same
location in an electric field. They are at
the same potential.
• However, the particle with the greater
charge has more potential energy.
• One key difference between charge and
mass: charge can also be negative.
Potential and the
Electric Field
Potential and the E Field
• Although we did not define V this way, it
is very closely related to E.
• Mathematically, this relationship is
lim
x  0
V
E 
x
• Generally, this expression is difficult to
evaluate without the aid of calculus.
Potential and the E Field
• However, we can see how V and E are
related for a constant electric field.
• Recall the following equations:
U  qVba
W  U
• So, the work done by the electric force is:
W  qVba

Potential and the E Field
• However, since the E field is constant
between the plates, we also know:
W  Fdcos
• Where here
F  qE
 0
• So
W  qEd
Potential and the E Field
• So, now we have two expressions for
work:
W  qVba W  qEd
• Setting these equal, we get:
Vba  Ed
Vba
E  
d
ONLY true if
the E field is
uniform
Example: Find the E Field
Two parallel plates are separated by a distance of 0.05 m
and are charged to produce a potential difference of 50 V
between them.
What is the magnitude of the electric field between the
plates?
50 V
-
+
0.05 m
Problems
• Do problems 5-8, 10, and 11 on page
489 of the book.
• Your homework is to finish these
problems.
A quick word about E
• Yesterday, we saw that a constant electric
field is related to potential by
Vba
E 
d
• In this equation, E has units of V/m. This is
an alternative way to express E, and it is
equivalent to the N/C formulation.

N
V
1 1
C
m
Potential of a Point Charge
Potential due to a Point Charge
• The potential a distance r away from a
point charge Q can be found from the
charge’s electric field using calculus.
• The result is
kQ
V 
r
Here, V = 0 at
r = .
Potential due to a Point Charge
• Notice that while E falls off
as 1/r2, V falls off as 1/r.
• The sign of the charge is
important.
– V is very large and positive
near a positive charge, and
decreases to zero far away.
– V is very large and negative
near a negative charge, and
increases to zero far away.
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Example: Potential of a Point
Charge
What is the potential 0.5 meters away from a point charge
magnitude +20 C?
Repeat the problem but make the charge -20 C.
Example: Bringing Two +
Charges Together
How much work is needed to bring a 3 C test charge from
a great distance away (say ) to a point 0.5 m away from a
+20 C charge?
Adding Potentials
• In the last unit, we learned that, to find
the net electric field, you add up the
individual fields caused by source
charges.
• Since E is a vector, you must use vector
addition.
• This is annoying.
Adding Potentials
• Luckily, potential is a scalar.
• This means you can add potentials just
like regular numbers, without worrying
about direction.
• You must include the sign of the source
charges however.
Example: Adding Potentials
Find the electric potential at points A and B due to the two
source charges shown.
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Problems
• Do problems 14, 15, 19, and 23 on
page 489.
• Your homework is to finish these
problems.
Problem Day
• Do problems 9, 13, and 16 on page
489.
• We will whiteboard these in 30 mins.
• If you finish early, work on problem 65
on page 491.
Whiteboarding Groups
Group
1
2
3
4
5
6
7
Members
John, Brie
Sarah, Drew, Angi
Kaleb, Jeremiah, Bailey
Miggy, Connor, Armen
Anthony, Abbey, Aidan
Rachel, Jacob, Piper
Robert, Ellen, Krystiana
Problem
14
15
19
23
9
13
16
Homework
• Read section 17-3
• Do problem 65 on page 491.
Equipotential Lines
Equipotential Lines
• Earlier in the unit, we likened potential
to “height” of the electric field.
• When we diagram heights in space, we
sometimes use a contour map.
• A contour line indicates an elevation
above sea level.
• Any point on that line is at that height.
Equipotential Lines
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Equipotential Lines
• We can do the same thing for the
electric field.
• Contour lines for the E field are called
equipotential lines.
• All points on an equipotential line are at
the same potential.
Equipotential Lines
• Since the work required to move a
charge between two points depends on
the voltage difference, no work is
needed to move a charge along an
equipotential line.
• No work is also needed to move a
charge perpendicular to the electric
field.
Equipotential Lines
• Therefore, equipotential lines must be
perpendicular to the E field at all
points.
• The E field always points towards lower
values of V.
• The closer together the lines are, the
stronger the electric field is in that area.
Equipotential Lines
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Equipotential Lines
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Potentials and Conductors
• Since the E field inside a conductor is
zero, the conductor must be entirely at
the same potential.
• A conductor is an equipotential.
• Incidentally, this is why E field lines are
always perpendicular to the surface of a
conductor.
Homework
• Read section 17-7
• Do problems 17 and 65 on page 489.
Capacitance and
Dielectrics
Capacitors
• A capacitor is a device used to store
electrical energy.
• They come in a variety of shapes and
sizes.
• Usually, a capacitor consists of two
sheets of metal separated by a small
distance.
Capacitors
• Capacitors have a wide range of uses,
from blocking electric surges, to camera
flashes, to computer RAM.
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Capacitors
• If a voltage difference is applied across
a capacitor (one plate is forced to be at
a higher potential than the other), the
two plates of the capacitor become
charged.
• One plate has a negative charge, the
other an equal amount of positive
charge.
Capacitors
• The amount of charge is proportional to
the voltage applied. In other words:
Q  CV
• The constant, C is called the
capacitance of the capacitor.
Capacitance
Q
C
V
Charge on
each plate
Applied
voltage
Capacitance
• The unit of capacitance is a coulombs
per volt.
• This unit is called a farad (F).
• Typical capacitors have capacitances in
the range of 1 pF (picofarads) to ~100
F (microfarads).
Parallel Plate Capacitors
• Capacitance, in general, does not
depend on Q or V.
• How much charge can fit on the
capacitor depends on the size of the
plates.
• The voltage difference depends on the
separation of the plates.
Parallel-Plate Capacitors
• We can use these observations to write
down the capacitance for a parallelplate capacitor with plates of area A,
separated by a distance d.
A
C  0
d
Area of each
plate
Parallel-plate
capacitor
Distance
only.
between plates
 0  8.85  10
12 C 2
2
N m
Example: Capacitance
A parallel-plate capacitor is made by placing two 20 cm x 3 cm
plates 1 mm apart.
a) What is the capacitance?
b) If the capacitor is connected to a 12-V battery, what is
the charge on each plate?
c) What is the magnitude of the E field between the
plates?
d) What would the area of the plates need to be to give
a capacitance of 1F?
Homework
• Do problems 33-39 odd on page 490.
• Check your answers in the back of the
book.
Circuit Symbols
• The capacitor is the first of many circuit
elements we will study this year.
• As it can be tedious to draw pictures of
actual circuits, we will be using circuit
diagrams to represent how circuits are
set up.
• Let’s go over some circuit symbols.
Circuit Symbols
• The simplest capacitor is just two
parallel plates separated by a small
distance.
• Capacitors are represented by the
symbol:
Circuit Symbols
• A battery is a device that maintains a
potential difference between two points
(terminals).
• We will discuss this more next unit.
• Batteries are represented with the
symbol:
+
Circuit Symbols
• A wire is a conductor used to connect
two circuit elements.
• Any two points on a wire are at the
same voltage.
• Wires are represented by lines in circuit
diagrams.
Circuit Symbols
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Dielectrics
Dielectrics
• The space inside most capacitors are is
not filled with air.
• Instead, the space is filled with an
insulating material.
• These materials are known as
dielectrics.
Dielectrics
•
There are two main reasons for using
a dielectric in a capacitor.
1. The plates can be put closer together,
because the dielectric keeps the charge
from jumping across the gap.
2. The dielectric itself increases the
capacitance.
Dielectrics
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Dielectrics
• When you place a dielectric in a
capacitor, the capacitance is increased
by a factor of K.
A
C  K 0
d
• K is called the dielectric constant of
the material.
• The
 value of K can be found in a table.
Dielectrics
• The formula for capacitance can also be
written:
A
C 
d
• Where
  K0
Permittivity of
the material.
Application: Computer Keyboards
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Homework
• Read sections 17-8 and 17-9.
• Do problems 42-45 on page 490.
• We will be having our next test this
Friday.
Whiteboarding Groups
Group
1
2
3
4
5
6
7
8
Members
John, Brie
Sarah, Angi
Kaleb, Jeremiah, Bailey
Miggy, Jacob
Anthony, Abbey, Aidan
Rachel, Piper, Drew
Connor, Armen
Robert, Ellen, Krystiana
Problem
42
43
44
45
33
35
37
39
Dielectric Breakdown
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Dielectric Breakdown
• We learned earlier this year that
dielectrics (insulators) prevent charge
from flowing.
• This is only true below if the voltage
difference across the material is below a
certain level.
• If the voltage is too high, the material
becomes ionized and charge is able to
flow.
Dielectric Breakdown
• To illustrate this, we will use the
phenomenon of lightning.
• During a storm, clouds acquire a net
negative charge.
• The process by which this happens is
not agreed upon by all scientists.
Dielectric Breakdown
• The Earth is a conductor and the
negative charge of the cloud repels
electrons from the surface of the Earth.
• The surface has a net positive charge.
• This leads to an electric field similar to
that of a parallel-plate capacitor.
Dielectric Breakdown
• The voltage difference between the
cloud and the Earth creates a force
attracting the electrons towards the
Earth.
• The electrons are prevented from
flowing by the air, which is an insulator.
Dielectric Breakdown
• However, if the voltage is large
(~10,000 volts/in2 for air), the force is
strong enough to pull electrons away
from the air molecules.
• The air becomes ionized.
• Ions are excellent conductors.
Dielectric Breakdown
• The electrons in the cloud now have a clear
conductive path and discharge into the
ground.
• This reduces the potential difference between
cloud and ground.
• The voltage is no longer strong enough to
ionize the air, and no more charge is able to
flow until the charge builds up again in the
cloud.
Dielectric Breakdown
• Dielectric breakdown can occur in any
insulating material.
• The voltage needed for breakdown to
occur depends on the material.
• For this reason, capacitors are rated for
a maximum voltage.
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Storage of
Electrical Energy
Storage of Electrical Energy
• We defined a capacitor as a device for storing
electrical energy.
• We now want to describe how much energy is
stored in the capacitor.
• If a capacitor is discharged, the charge flows
off the plates and can do work (in an electric
circuit).
• Therefore, this is potential energy.
Storage of Electrical Energy
U  QV
1
2
U  CV
1
2
2
Q
U
C
1
2
Q - Charge on each
plate of the capacitor.
2
V - Potential difference
across the capacitor.
C - Capacitance of the
capacitor.
Storage of Electrical Energy
• It is interesting to note that the energy in
these equations is stored in the electric
field between the plates of the capacitor.
• This suggests that any electric field
contains potential energy.
Storage of Electrical Energy
• For a regular shape (like a parallelplate) capacitor, it is fairly easy to
quantify the amount energy stored in
the field.
U   0 E Ad
1
2
2
• This is only true for a parallel-plate
capacitor.
• See section 17-9 for the derivation.
Storage of Electrical Energy
• However, it is usually more useful to talk
about energy density.
• This is the amount of energy stored in
the field per unit volume.
energy density   0 E
1
2
• This is true for any electric field.
2
Homework
• Do problems 46-49 on page 490.
• You may want to refer to section 17-9
while doing these problems.
Announcements
• We will be having our next test this Friday.
• I will give you a formula sheet on the test.
• The AP review sessions begin this Friday
after class.
• I will start Tuesday morning tutoring next
week.
Problem Day
• Do problems 60, 61, 67, 70, 73, and 75
on page 491-492.
• Your homework is to finish these
problems.
Whiteboarding Groups
Group
Members
Problem
1
John, Angi, Ellen, Krystiana
60
2
Brie, Armen, Aidan
61
3
Kaleb, Jeremiah, Bailey
67
4
Miggy, Sarah, Abbey
70
5
Anthony, Drew, Connor
73
6
Rachel, Jacob, Piper,
75
Robert
The Electrocardiogram
The Heart
• The human heart is responsible to
pumping blood to the rest of the body.
• It is electrical impulses that make the
heart beat.
• These electrical impulses can be
detected by an electrocardiogram
(EKG).
The Heart
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The Heart
• When the heart is at
rest (in between
beats), there is a
natural separation of
charge between the
inside of heart cells
and the outside.
• This separation of
charge leads to a
potential difference
across the cell
membrane.
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The Heart
• The inside of a heart
cell is at a potential
that is about 90 mV
less than the outside of
the cell.
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• When the muscle
contracts, the
membrane allows the
positive ions to pass
inside the cell.
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The Heart
• This “depolarization” of
the cell starts at one
side and progresses
across the cell.
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• The charge distribution
of the cell has now
changed.
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• This means the
voltage difference has
also changed.
The Heart
• Once the cell is
depolarized, the
process reverses and
the cell re-polarizes to
its original state.
• The changes in
voltage are read by the
EKG and analyzed to
determine the health of
the heart.
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The EKG
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• The EKG graphs the voltage between
two points on opposite sides of the
heart as a function of time.
• A typical EKG graph is consists of three
“waves” that represent a single
heartbeat.
The EKG
P wave
QRS complex
T wave
Contraction of
the atria
Contraction of
the ventricles
Re-polarization
of the heart
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