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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 required next Monday, memos due next Wednesday.

Are electric fields getting you down?

Can’t sleep at night because visions of electric fields dance in your head?

Do you keep turning around and looking behind you because you are afraid an electric field might be following you?

Are you ready to try something new?

http://www.aps.org/units/dcmp/gallery/images/electricfield.jpg

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.

f i conservative i fU U U W

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 external conservativei f i f

W W

Always ask yourself which work you are calculating.

*for example, if you “slowly” lift a book, or “slowly” push two negatively charged balloons together

This definition is from Physics 1135.

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

1 2 1 212

12 0 12

q q q q1U r k .

r 4 r

Sooner or later I am going to forget and put in a 1/r2 dependence. Don’t be bad like me.

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.

Still Another Important Fact.

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?”

1 2 1 212

12 0 12

q q q q1U r k

r 4 r

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 two protons separated by a typical proton-proton intranuclear distance of 2x10-15 m.

+1.15x10-13 J

What is the meaning of the + sign in the result?

+e+e

D=2x10-15 m

19 19

91 215

12

1.6 10 1.6 10e eq qU k k 9 10

r D 2 10

13U 1.15 10 J

Example: calculate the electric potential energy of a hydrogen atom (electron-proton distance is 5.29x10-11 m). -4.36x10-18 J


Example: calculate the electric potential energy of a hydrogen atom (electron-proton distance is 5.29x10-11 m). -4.36x10-18 J

What is the meaning of the - sign in the result? Is that a small energy? I’ll have more to say about the energy at the end of the lecture.

-e+e

D=5.29x10-11 m

19 19

91 211

12

1.6 10 1.6 10e eq qU k k 9 10

r D 5.29 10

18U 4.36 10 J

Today’s agenda:

Electric potential energy (continued).You must be able to use electric potential energy in work-energy calculations.




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)if. The grav-itational force does + work.

y

graphic “borrowed” from http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html

What force does Wc? Force due to gravity.

x

Ui = mgyi

Uf = 0

yi

An object of mass m in a gravitational field has potential energy U(y) = mgy and “feels” a gravitational force FG = GmM/r2, attractive.

Remember conservation of energy from Physics 23?

http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html

+ + + + + + + + + + + + + +

- - - - - - - - - - - - - - - - - - -

+

E

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.

F

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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?

f i other i fE E W

“The hammer equation.”—©Prof. R. J. Bieniek

http://etc.usf.edu/clipart/index.htm

Here is another important Physics 1135 Starting Equation, which you may need for tomorrow’s homework…

The Work-Energy Theorem:

net i fW 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]ba

Example: two isolated protons are constrained to be a distanceD = 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?

2.63x104 m/s

To be worked at the blackboard in lecture…

2.63x104 m/s

Skip to slide 24.


2.63x104 m/sWe need to do some thinking first.

What is the proton’s potential energy when they reach their maximum speed?

How far apart are the protons when they reach their maximum speed?


2.63x104 m/s+e+e

ri=2x10-10 m

v=0 v=0Initial

+e+e

rf=v v

Final

There is an unasked conservation of momentum problem buried in here, isn’t there!

+e+e

ri=2x10-10 m

v=0 v=0Initial

+e+e

rf=v v

Final

f i other i fE E W

f f i i other i fK U K U W

0 0 0

if UK

How many objects are moving in the final state?

+e+e

ri=2x10-10 m

v=0 v=0Initial

+e+e

rf=v v

Final

if UK

Two.

How many Kf terms are there? Two.

How many pairs of charged particles in the initial state?

One.

How many Ui terms are there? One.

+e+e

ri=2x10-10 m

v=0 v=0Initial

+e+e

rf=v v

Final

if UK

i

2p r

eekvm

2

12

s

m1063.2

102 1067.1

106.1 109

rm

ke v 4

10-27-

219-9

ip

2

Is that fast, or slow?

E Ef Ei E i fU U U W

f f

i i

r r1 2

E E 2i f r r12

k q qW F d dr

r

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.

The subscript “E” is to remind you I am talking about electric potential energy. After this slide, I will drop the subscript “E.”

Another way to calculate electrical potential energy.

Move one of charges from ri to rf, in the presence the other charge.

f f

i i

r r f2

E 1 1 2 1 22r r i12

kqU q dr q E dr q E d

r

Move q1 from ri to rf, in the presence of

q2.

f

i

r1 22r

12

k q q dr

r

A justification, but not a mathematically “legal” derivation.

Generalizing:

f

f i iU U q E d

“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.

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.”

So far in today’s lecture…

I reminded you of some energy concepts from Physics 1135:

f i conservative i fU U U W

external conservativei f i fW W

definition of potential energy

true if kinetic energy is constant

f i other i fE E W

everybody’s favorite Phys.

1135 equation

net i fW K

work-energy theorem

You mastered all of the above equations in Physics 1135.

So far in today’s lecture…

Then I “derived” an equation for the electrical potential energy of two point charges

1 2 1 212

12 0 12

q q q q1U r k .

r 4 r

f

f i iU U q E d

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

Above is today’s stuff. So far. Lots of lecturing for only two equations.

Today’s agenda:





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:

0

0

q 00

FE = lim

q

Similarly, it is useful to define the potential in terms of the potential energy of a test charge q0:

0q 0

0

U rV r = lim

q

The electric potential V is independent of the test charge q0.

Later you’ll get an Official Starting Equation version of

this.

A point in space can have an electric potential even if there is no charge around to “feel” it.

so that the electric potential of a point charge q is

1 2 2

1 1 0 12 0 12

U r q q q1 1 1V(r)

q q 4 r 4 r

0

1 qV r .

4 r

The electric potential difference between points a and b is b

b ba

a a

r

E r rr E

r r0 0 0

F d FUV d E d .

q q q

f

iV E d

q1 is the test charge, q2 is the charge that

gives rise to the potential V(r) that q1 “feels.” (q1 probes

the potential)

Sooner or later I am going to forget and put in a 1/r2 dependence. Don’t be bad like me.

Only valid for a point charge!

E is likely due to a collection of point charges.

electric potential of a point charge

One more starting equation

f i

UV V V

q

*Very Handy Version: U q V A particle of charge q moved through a potential difference V gains (or loses) potential energy q V.

b

b ba

a a

r

E r rr E

r r0 0 0

F d FUV d E d .

q q q

Drop the subscript on the q0. It was there to remind us that q0 is the charge

that “feels” the potential.

Copied from previous slide.

*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.

0

U rV r = .

q

The units of potential are joules/coulomb:

1 joule1 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.

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.

Example: a electron is released in a region in space where there is an electric potential. Describe the subsequent motion of the electron.

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


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 v v

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

+ -

D

+e -eP VP?

V 27.2

105.29

101.6 109

D

ek

r

kq V

11

99

p

In the second part of today’s lecture…

I defined electric potential as potential energy per unit of charge

…and found the potential due to a point charge…

This equation also lets you calculate the change in potential energy when a charge q moves through a potential difference V.

U q V

0

1 qV r

4 r

…and showed how to calculate the potential difference between two points in an electric field

f

iV E d .

We haven’t used this yet, but will

eventually.

Today’s agenda:





Electric Potential Energy of a System of Charges

Electric Potential 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!)

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!)

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:

before

after

q1

U 0

q1 q2

1 2q qU k

r

r

Example: electric potential energy of three charged particles

To find the electric potential energy for a system of three charges, we bring a third charge in from an infinite distance away:

before

q1

after

q1 q2

1 3 2 31 2

12 13 23

q q q qq qU k

r r r

r12

q2

r12

1 2

12

q qU k

r

q3

r13 r23

We have to add the potential energies of each pair of charged particles.

Electric Potential of a Charge Distribution (details next lecture)

Collection of charges: iP

i0 i

q1V .

4 r

Charge distribution:

P is the point at which V is to be calculated, and ri is the distance of the ith charge from P.

Pr

dq

0

1 dqV .

4 r

Potential at point P.We’ll work with this next

lecture.

“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?”

Exception: in 23.5 (if assigned), because the spheres are “small” you can calculate potential energies using the equation for the potential energy of point charges.

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.

q2q1

3 m

P

4 mx

yi 1 2

Pi i 1 2

-6 -69

3

q q qV = k = k +

r r r

1×10 -4×10= 9×10 +

3 5

= - 4.2×10 V

r1r2

Example: how much work is required to bring a +3 C point charge from infinity to point P? (And what assumption must we make?)

q2q1

3 m

P

4 mx

y

q3

external 3W U q V

external 3 PW q V V

externalW E K U 0

0

6 3externalW 3 10 4.2 10

3externalW 1.26 10 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.

q2q1

3 m

P

4 mx

y

q31 2 1 3 2 3

12 13 23

q q q q q qU = k + +

r r r

-6 -6 -6 -6 -6 -6

91×10 -4×10 1×10 3×10 -4×10 3×10

U = 9 10 + +4 3 5

-2U = - 2.16 10 J

r13r23

r12

Today’s agenda:





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

-191 eV= 1.6 10 C 1 V

-191 eV= 1.6 10 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?

-18 -18-19

1 eVU = -4.36 10 J = -4.36 10 J -27.2 eV

1.6 10 J

“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.

Remember your Physics 1135 hammer equation?

f i other i fE E W

What “goes into” Ef and Ei? What “goes into” Wother?

Homework Hints!

You’ll need to use starting equations from Physics 1135!

f i c i fU U W

This is also handy:

“Potential of a with respect to b” means Va - Vb

Homework Hints!

Work-Energy Theorem:

net i fW K

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