Transcript Slide 1

http://www.nearingzero.net (nz136.jpg)
If there’s toast in the toaster and no one sees it, is there really toast in the toaster? Check with your local quantum physicist before you answer!
Announcements
Reminder: if you have not yet done so, provide me the
necessary information about your Exam 1 special
circumstances (late exam, test center accommodations,
official University event conflict). See lecture 4 for details.
Test center notification preferred this week, memos due next
week.
Today’s agenda:
Electric potential energy.
You must be able to use electric potential energy in work-energy calculations.
Electric potential.
You must be able to calculate the electric potential for a point charge, and use the electric
potential in work-energy calculations.
Electric potential and electric potential energy of a system of
charges.
You must be able to calculate both electric potential and electric potential energy for a
system of charged particles (point charges today, charge distributions next lecture).
The electron volt.
You must be able to use the electron volt as an alternative unit of energy.
Definition and Really Important fact to keep straight.
U  Uf  Ui   Wconservative if
This definition is
from Physics 23.
The change in potential energy is defined as the negative of the
work done by the conservative force which is associated with
the potential energy (today, the electric force).
If an external force moves an object “against” the conservative
force,* and the object’s kinetic energy remains constant, then
Wexternal if  Wconservative if
Always ask yourself which work you are calculating.
*for example, if you “slowly” lift a book, or “slowly” push two negatively charged balloons together
Another Important Fact.
Potential energies are defined relative to some configuration of
objects that you are free to choose.
For example, it often makes sense to define the gravitational potential energy of a ball to be
zero when it is resting on the surface of the earth, but you don’t have to make that choice.
“Available energy is the main object at stake in the
struggle for existence and the evolution of the
world.”—Ludwig Boltzmann
If I hold one proton in my right hand, and another proton in
my left hand, and let them go, they will fly apart. (You have to
pretend my hands are “physics” hands—they aren’t really there.)
“Flying” protons have kinetic energy, so when I held them at
rest, they must have had potential energy.
The electric potential energy of a system of two point charges
q1 and q2, separated by a distance r12 is
q1q 2
1 q1q 2
U  r12   k

.
r12
40 r12
This is not a definition; it is derived from the definition of potential energy.
Read your text, or ask me in the Learning Center where this comes from.
Sooner or later I am
going to forget and put
in a 1/r2 dependence.
Don’t be bad like me.
Still Another Important Fact.
q1q 2
1 q1q 2
U  r12   k

r12
40 r12
Our equation for the electric potential energy of two charged
particles uses the convention that the potential energy is zero
when the particles are infinitely far apart.
Does that make sense?
It’s the convention you must use if you want to use the equation for potential energy of point
charges! If you use the above equation, you are “automatically” using this convention.
Homework hint: if charged particles are “far” apart, their potential energy is zero. So how far is “far?”
Example: calculate the electric potential energy of two protons
separated by a typical proton-proton intranuclear distance of
2x10-15 m.
+1.15x10
-13 J
To be worked at the blackboard in lecture.
Example: calculate the electric potential energy of a hydrogen
atom (electron-proton distance is 5.29x10-11 m).
-4.36x10
-18
To be worked at the blackboard in lecture.
J
Today’s agenda:
Electric potential energy (continued).
You must be able to use electric potential energy in work-energy calculations.
Electric potential.
You must be able to calculate the electric potential for a point charge, and use the electric
potential in work-energy calculations.
Electric potential and electric potential energy of a system of
charges.
You must be able to calculate both electric potential and electric potential energy for a
system of charged particles (point charges today, charge distributions next lecture).
The electron volt.
You must be able to use the electron volt as an alternative unit of energy.
Remember conservation of energy from Physics 23?
An object of mass m in a gravitational field has potential
energy U(y) = mgy and “feels” a gravitational force FG =
GmM/r2, attractive.
y
If released, it gains kinetic
energy and loses potential
energy, but mechanical energy
is conserved: Ef=Ei. The
change in potential energy is
Uf - Ui = -(Wc)if. The gravitational force does + work.
Ui = mgyi
yi
x
Uf = 0
What force does Wc? Force due to gravity.
graphic “borrowed” from http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html
A charged particle in an electric
field has electric potential
energy.
++++++++++++++
It “feels” a force (as given by
Coulomb’s law).
It gains kinetic energy and loses
potential energy if released. The
Coulomb force does positive
work, and mechanical energy is
conserved.
+
E
F
-------------------
Now your deep philosophical question for the day…
If you have a great big nail to drive, are you going
to pound it with a dinky little screwdriver?
Or a
hammer?
Ef  Ei   Wother if
“The hammer equation.”—©Prof. R. J. Bieniek
Here is another important Physics 23 Starting Equation, which
you may need for tomorrow’s homework…
The Work-Energy Theorem:
Wnet if  K
Wnet is the total work, and includes work done by the
conservative force (if any) and all other forces (if any).
Notation: Wab = Wa-Wb = [W]ba
Example: two isolated protons are constrained to be a distance
D = 2x10-10 meters apart (a typical atom-atom distance in a
solid). If the protons are released from rest, what maximum
speed do they achieve, and how far apart are they when they
reach this maximum speed?
2.63x10 m/s
4
To be worked at the blackboard in lecture…
2.63x104 m/s
Another way to calculate electrical potential energy.
UE  UEf  UEi   WE if
  WE if    FE  d   
rf
rf
ri
ri
k q1q 2
dr
2
r12
The subscript “E” is to
remind you I am talking
about electric potential
energy. After this slide, I
will drop the subscript “E.”
Move one of charges from
ri to rf, in the presence the
other charge.
The minus sign in this equation comes from the definition of change in potential energy. The sign from the dot product is
“automatically” correct if you include the signs of q and q0.
U E  q1 
rf
ri
rf
f
kq 2
dr  q1  E 2dr  q1  E 2  d
2
ri
i
r12
Move q1 from ri to rf, in
the presence of q2.
A justification, but not a mathematically “legal” derivation.
Generalizing:
f
Uf  Ui  q  E  d
i
When a charge q is moved from one position to another in the presence of
an electric field due to one or more other charged particles, its change in
potential energy is given by the above equation.
I’ve done something important here. I’ve generalized from the
specific case of one charged particle moving in the presence of
another, to a charged particle moving in the electric field due to
all the other charged particles in its “universe.”
“i” and “f” refer to the two points for which we are calculating the potential energy difference. You could also
use “a” and “b” like your text does, or “0” and “1” or anything else convenient. I use “i” and “f” because I
always remember that (anything) = (anything)f – (anything)i.
So far in today’s lecture…
I reminded you of some energy concepts from Physics 23:
U  Uf  Ui   Wconservative if
definition of potential energy
Wexternal if  Wconservative if
true if kinetic energy is constant
Ef  Ei   Wother if
Wnet if  K
everybody’s favorite Phys. 23
equation
work-energy theorem
You mastered all of the above equations in Physics 23.
So far in today’s lecture…
Then I “derived” an equation for the electrical potential
energy of two point charges
q1q 2
1 q1q 2
U  r12   k

.
r12
40 r12
I also derived an equation (which we haven’t used yet) for
the change in electrical potential energy of a point charge
that moves in the presence of an electric field
f
Uf  Ui  q  E  d
i
Above is today’s stuff. So far. Lots of lecturing for only two equations.
Today’s agenda:
Electric potential energy.
You must be able to use electric potential energy in work-energy calculations.
Electric potential.
You must be able to calculate the electric potential for a point charge, and use the electric
potential in work-energy calculations.
Electric potential and electric potential energy of a system of
charges.
You must be able to calculate both electric potential and electric potential energy for a
system of charged particles (point charges today, charge distributions next lecture).
The electron volt.
You must be able to use the electron volt as an alternative unit of energy.
Now I’m going to do something different, and introduce the
“electric potential.”
Electric potential energy is “just like” gravitational potential
energy.
Except that all matter exerts an attractive gravitational force, but charged particles exert
either attractive or repulsive electrical forces—so we need to be careful with our signs.
Electric potential is the electric potential energy per unit of
charge.
Electric Potential
In lecture 1 we defined the electric field by the force it exerts
on a test charge q0:
F0
E = lim
q0 0 q
0
Similarly, it is useful to define the potential in terms of the
potential energy of a test charge q0:
Ur 
V  r  = lim
q0 0
q0
Later you’ll get an Official
Starting Equation version of this.
The electric potential V is independent of the test charge q0.
A point in space can have an electric potential even if there is no charge around to “feel” it.
Ur
1 1 q1q 2
1 q2
V(r) 


q1
q1 40 r12
40 r12
q1 is the test charge, q2
is the charge that gives
rise to the potential V(r)
that q1 “feels.” (q1
probes the potential)
so that the electric potential of a point charge q is
Only valid for a
point charge!
1 q
V r 
.
40 r
Sooner or later I am
going to forget and put
in a 1/r2 dependence.
Don’t be bad like me.
electric potential of a point charge
The electric potential difference between points a and b is
rb
rb F
rb
U ra FE  d
E
V 

 
 d   E  d .
ra q
ra
q0
q0
0
E is likely due to
a collection of
point charges.
f
V    E  d
i
One more starting equation
rb
rb F
rb
U ra FE  d
E
V 

 
 d   E  d .
ra q
ra
q0
q0
0
U
V  Vf  Vi 
q
*Very Handy Version: U  q V
Copied from
previous slide.
Drop the subscript on the
q0. It was there to remind us
that q0 is the charge that
“feels” the potential.
A particle of charge q moved through a
potential difference V gains (or loses)
potential energy q V.
*In other words, you usually start with this version of the equation.
Things to remember about electric potential:
 Electric potential and electric potential energy are related, but
not the same.
Electric potential difference is the work per unit of charge
that must be done to move a charge from one point to
another without changing its kinetic energy.
 The terms “electric potential” and “potential” are used
interchangeably.
Ur 
.
 The units of potential are joules/coulomb: V  r  =
q0
1 joule
1 volt =
1 coulomb
Things to remember about electric potential:
 Only differences in electric potential and electric potential
energy are meaningful.
It is always necessary to define where U and V are zero. In
this lecture we define V to be zero at an infinite distance
from the sources of the electric field.
Sometimes (e.g., circuits) it is convenient to define V to be
zero at the earth (ground).
It will be clear from the context where V is defined to be
zero. Most equations for this chapter assume V=0 at infinite
separation of charges.
Saying “take V to be zero when the charges are far apart means “it’s OK to use the equations in this chapter.”
Two conceptual examples.
Example: a proton is released in a region in space where there
is an electric potential. Describe the subsequent motion of the
proton.
The proton will move towards the region of lower potential. As it moves, its
potential energy will decrease, and its kinetic energy and speed will increase.
Example: a electron is released in a region in space where there
is an electric potential. Describe the subsequent motion of the
electron.
The electron will move towards the region of higher potential. As it moves,
its potential energy will decrease, and its kinetic energy and speed will
increase.
Protons fall down, electrons fall up.
What is the potential due to the proton in the hydrogen atom at
the electron’s position (5.29x10-11 m away from the proton)?
27.2V
To be worked at the blackboard in lecture.
Important note:
V
this is the symbol for electrical potential
V
this is the symbol for the unit (volts) of electrical
potential
v
this is the symbol for magnitude of velocity, or speed
Don’t get your v’s and V’s mixed up! Hint: write your speed v’s
as script v’s, like this (or however you want to clearly indicate a
lowercase v):
v 
vv
In the second part of today’s lecture…
I defined electric potential as potential energy per unit of
charge
U  q V
This equation also lets you calculate the
change in potential energy when a charge
q moves through a potential difference V.
…and found the potential due to a point charge…
1 q
V r 
40 r
…and showed how to calculate the potential difference
between two points in an electric field
f
V   E  d .
i
We haven’t used this yet, but will eventually.
Today’s agenda:
Electric potential energy.
You must be able to use electric potential energy in work-energy calculations.
Electric potential.
You must be able to calculate the electric potential for a point charge, and use the electric
potential in work-energy calculations.
Electric potential and electric potential energy of a
system of charges.
You must be able to calculate both electric potential and electric potential energy for a
system of charged particles (point charges today, charge distributions next lecture).
The electron volt.
You must be able to use the electron volt as an alternative unit of energy.
Electric Potential Energy of a System of Charges
Electric potential energy comes from the interaction between
pairs of charged particles, so you have to add the potential
energies of each pair of charged particles in the system.
(Could be a pain to calculate!)
Electric Potential of a System of Charges
The potential due to a particle depends only on the charge of
that particle and where it is relative to some reference point.
The electric potential of a system of charges is simply the sum
of the potential of each charge. (Much easier to calculate!)
Example: electric potential energy of three charged particles
A single charged particle has no electrical potential energy. To
find the electric potential energy for a system of two charges,
we bring a second charge in from an infinite distance away:
r
q1
q1
U 0
before
q2
q1q 2
U k
r
after
To find the electric potential energy for a system of three
charges, we bring a third charge in from an infinite distance
away:
r12
q1
q2
before
q1q 2
U k
r12
q1
r12
q2
r13
r23
q3
after
 q1q 2 q1q3 q 2q 3 
U  k



r13
r23 
 r12
We have to add the potential energies of
each pair of charged particles.
Electric Potential of a Charge Distribution
1
Collection of charges: VP 
40
(details next lecture)
qi
i r .
i
P is the point at which V is to be calculated, and ri is the distance of the ith
charge from P.
Charge distribution:
1
dq
V
.

40 r
Potential at point P.
We’ll work with this next lecture.
dq
P
r
“In homework and on exams, can I automatically assume the
electric potential outside of a spherically-symmetric charge
distribution with total charge Q is the same as the electric potential
of a point charge Q located at the center of the sphere?”
Example: a 1 C point charge is located at the origin and a -4
C point charge 4 meters along the +x axis. Calculate the
electric potential at a point P, 3 meters along the +y axis.
y
 q1 q 2 
qi
VP = k  = k  + 
i ri
 r1 r2 
-6
-6


1×10
-4×10
9
= 9×10 
+

3
5


P
3m r
1
q1
r2
4m
q2
x
= - 4.2×103 V
Example: how much work is required to bring a +3 C point
charge from infinity to point P?
(And what assumption must we make?)
0
y
Wexternal  E  K  U
q3
Wexternal  U  q3V
P
0
3m
q1
Wexternal  q3  VP  V 
4m
q2
x
Wexternal  3 106  4.2 103 
Wexternal  1.26 103 J
The work done by the external force was negative, so the work done by the electric field was
positive. The electric field “pulled” q3 in (keep in mind q2 is 4 times q1).
Positive work would have to be done by an external force to remove q3 from P.
Example: find the total potential energy of the system of three
charges.
y
q3
 q1 q 2 q1 q 3 q 2 q 3 
U = k
+
+

r
r
r
13
23 
 12
P
r23
3m r
13
r12
q2
4m
q1


x
 

 

-6
-6
-6
-6
 1×10-6 -4×10-6
1×10
3×10
-4×10
3×10
U = 9 109 
+
+

4
3
5

U = - 2.16 10-2 J
 


Today’s agenda:
Electric potential energy.
You must be able to use electric potential energy in work-energy calculations.
Electric potential.
You must be able to calculate the electric potential for a point charge, and use the electric
potential in work-energy calculations.
Electric potential and electric potential energy of a system of
charges.
You must be able to calculate both electric potential and electric potential energy for a
system of charged particles (point charges today, charge distributions next lecture).
The electron volt.
You must be able to use the electron volt as an alternative unit of energy.
The Electron Volt
An electron volt (eV) is the energy acquired by a particle of
charge e when it moves through a potential difference of 1 volt.
U= qV
1 eV= 1.6 10-19C  1 V 
1 eV= 1.6 10-19 J
This is a very small amount of energy on a macroscopic scale,
but electrons in atoms typically have a few eV (10’s to 1000’s)
of energy.
Example: on slide 9 we found that the potential energy of the
hydrogen atom is about -4.36x10-18 joules. How many electron
volts is that?
 1 eV 
U = -4.36 10 J =  -4.36 10 J  
 -27.2 eV
-19 
 1.6 10 J 
-18
-18
“Hold it! I learned in Chemistry (or high school physics) that the
ground-state energy of the hydrogen atom is -13.6 eV. Did you
make a physics mistake?”
The ground-state energy of the hydrogen atom includes the
positive kinetic energy of the electron, which happens to have a
magnitude of half the potential energy. Add KE+PE to get
ground state energy.
Homework Hints!
You’ll need to use starting equations from Physics 23!
Remember your Physics 23 hammer equation?
Ef  Ei   Wother if
What “goes into” Ef and Ei? What “goes into” Wother?
This is also handy:
Uf  Ui   Wc if
Homework Hints!
Work-Energy Theorem:
Wnet if  K
“Potential of a with respect to b” means Va - Vb