Lecture05 Electric Potential; Electric Potential Energy

8/3/2019 Lecture05 Electric Potential; Electric Potential Energy

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http://www.nearingzero.net(nz136.jpg)

If theres 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!


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Announcements

Reminder: if you have not yet done so, provide me thenecessary information about your Exam 1 specialrequirements (late exam, test center accommodations,official University event conflict). See lecture 4 for details.

Test center notification required this week, memos due nextweek.

E

E dA* !

&&

is the definition of electric flux through asurface.

If appropriate, you should use or one of theother simpler versions. I wont make that an OSE because it is not always valid.

EE A* !

&&


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Announcements (continued)

If you are going to miss recitation, let your recitationinstructor know, but thatwill not necessarily excuse youfrom boardwork and will not excuse you fromturning in homework.

Find a way to get your homework to your recitationinstructorThe above was presented in the contextof a temporarily debilitating illness and is not ablanket excuse to skip recitation, and isabsolutely not authorization to turn in all your

homework electronically. You are expected toattend recitation!


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Todays 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 ofcharges.You must be able to calculate both electric potential and electric potential energy for asystem 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.


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Definition and Really Important fact to keep straight.

? Af i conservative i fU U U W p( ! !

The change in potential energy is the negative of the work doneby the conservative force which is associated with the potentialenergy (today, the electric force).

If an external force moves an object against the conservativeforce,* and the objects kinetic energy remains constant, then

? A ? Aexternal conservativei f i f W Wp p!

Always ask yourself which work you are calculating.

*for example, if you slowly pick up a book, or slowly push two negatively charged balloons together

This definition isfrom Physics 23.


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Another Important Fact.

Potential energies are defined relative to some configuration ofobjects that you are free to choose.

For example, it often makes sense to define the gravitational potential energy of a ball to bezero when it is resting on the surface of the earth, but you donthave to make that choice.


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If I hold one proton in my right hand, and another proton inmy left hand, and let them go, they will fly apart. (You have to

pretend my hands are physics handsthey arent really there.)

Flying protons have kinetic energy, so when I held them atrest, they must have had potential energy.

The electric potential energy of a system of two point chargesq1 and q2, separated by a distance r12 is

1 2 1 21212 0 12

q q q q1U r k .

r 4 r! !

TI

Sooner or later I amgoing to forget and putin a 1/r2 dependence.Dont be bad like me.

This is not a definition; it is derived from the definition of potential energy.

The next slide, which I will skip in lecture, shows where this comes from. Click here to skip the next slide.


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Where does come from?

When we have a conservative force (amount of work done by the force isindependent of path taken) we can define a potential energy.

In Physics 23, you defined potential energy like this:

1 2

12

q qU k

r

!

? Af i C i fU U U W p( ! !

f f

i i

r r1 2

E E 2r r12

kq qW F d dr

r! !

&&"

We can show the electric force is a conservative force. Then using

Coulombs law, we write

and do the integral to derive the expression for U. (We move one of the protons fromri to rfand calculate the work to do that.)

Explanatory slide, for those who like to know where things come from.

You may need to use this inhomework for tomorrow!


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Still Another Important Fact.

Potential energies are defined relative to some configuration ofobjects that you are free to choose.

Our equation for the electric potential energy of two chargedparticles uses the convention that the potential energy is zero

when the particles are infinitely far apart.

For example, it often makes sense to define the gravitational potential energy of a ball to bezero when it is resting on the surface of the earth, but you donthave to make that choice.

I told you the stuff above several slides back.

Does that make sense?

Its the convention you must use if you want to use the equation for potential energy of pointcharges! If you use that equation, you are automatically using this convention.

Homework hint: if charged particles are initially far apart, their initial potential energy is zero. So how far is far?


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Example: calculate the electric potential energy of two protonsseparated 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?

Example: calculate the electric potential energy of a hydrogenatom (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 smallenergy? Ill have more to say about the energy at the end ofthe lecture.


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If released, it gains kinetic

energy and loses potentialenergy, but mechanical energyis conserved: Ef=Ei. Thechange in potential energy isUf- Ui = -(Wc)ipf. 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= mgy

i

Uf = 0

yi

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

Remember conservation of energy from Physics 23?


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

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

+

E

A charged particle in an electricfield has electric potentialenergy.

It feels a force (as given by

Coulombs law).

It gains kinetic energy and losespotential energy if released. The

Coulomb force does positivework, and mechanical energy isconserved.

F


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Now your deep philosophical question for the day


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If you have a great big nail to drive, are you goingto pound it with a dinky little screwdriver?

Or a hammer?

? Af i other i fE E W p !

The hammer equation.Prof. R. J. Bieniek


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Numerical example.

Two isolated protons are constrained to be a distanceD = 2x10-10 meters apart (a typical atom-atom distance in asolid). If the protons are released from rest, what maximumspeed do they achieve, and how far apart are they when they

reach this maximum speed? 2.63x104

m/s

To be worked at the board.

2.63x104

m/s


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? AE Ef Ei E i fU U U W p( ! !

? Af f

i i

r r1 2

E E 2i fr r 12

kq qW F d dr

rp

! ! &&"

The minus sign in this equation comes from the definition of change in potential energy. The sign from the dot product isautomatically correct if you include the signs of q and q0.

The subscript E is toremind you I am talkingabout electric potentialenergy. After this slide, Iwill drop the subscript E.

Another way to calculate electrical potential energy.

Move one of charges fromri 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, inthe presence of q2.

f

i

r1 2

2r 12

kq qdr

r

!


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Generalizing:

f

f ii

U U q E d ! &&"

i and f refer to the two points for which we are calculating the potential energy difference. You couldalso 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 inpotential energy is given by the above equation.

Ive done something important here. Ive generalized from thespecific case of one charged particle moving in the presence ofanother, to a charged particle moving in the electric field due toall the other charged particles in its universe.


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Quiz time (maybe for points, maybe just for practice!)


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Todays agenda:







21/41

Now Im going to do something different, and introduce theelectric potential.

Electric potential energy is just like gravitational potentialenergy.

Except that all matter exerts an attractive gravitational force, but charged particles exert

either attractive or repulsive electrical forcesso we need to be careful with our signs.

Electric potential is the electric potential energy per unit ofcharge.


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Electric Potential

In lecture 2 we defined the electric field by the force it exertson a test charge q0:

0

0

q 00

FE = lim

qp

&&

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

0q 0

0

U rV r = limqp

&

&

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

Later youll get an OfficialStarting Equation version of this.

Apoint in space can have an electric potential even if there is no charge around to feel it.


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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

! ! !

TI TI

0

1 qV r .

4 r

!

TI

The electric potential difference between points a and b isb

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, q2is the charge that givesrise to the potential that

q1 feels. (q1 probesthe potential)

Sooner or later I amgoing to forget and putin a 1/r2 dependence.

Dont be bad like me.

Only valid for apoint charge!

E is likely due toa collection ofpoint charges.

electric potential of a point charge


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One more starting equation

f i

UV V V

q

(( ! !

Very Handy Version: U q V( ! (Aparticle of charge q moved through apotential 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 theq0. It was there to remind us

that q0 is the charge thatfeels the potential.

Copied fromprevious slide.


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Things to remember about electric potential:

y Electric potential and electric potential energy are related, butnot the same.

Electric potential difference is the work per unit of chargethat must be done to move a charge from one point to

another without changing its kinetic energy.

y The terms electric potential and potential are usedinterchangeably.

0

U rV r = .q

&

&

y The units of potential are joules/coulomb:

1 joule1 volt =

1 coulomb


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Things to remember about electric potential:

y Only differences in electric potential and electric potentialenergy are meaningful.

It is always necessary to define where U and V are zero. Inthis 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 bezero at the earth (ground).

It will be clear from the context where V is defined to bezero, and I do not foresee you experiencing any confusionabout where V is zero. Most equations for this chapterassume V=0 at infinite separation of charges.

Saying take V to be zero when the charges are far apart means its OK to use the equations in this chapter.


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Two conceptual examples.

Example: a proton is released in a region in space where thereis an electric potential. Describe the subsequent motion of theproton.

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

the electron.

The proton will move towards the region of lower potential. As it moves, its

potential energy will decrease, and its kinetic energy and speed willincrease.

The electron will move towards the region of higher potential. As it moves,its potential energy will decrease, and its kinetic energy and speed willincrease.


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Simple numerical example.

What is the potential due to the proton in the hydrogen atom atthe electrons position (5.29x10-11 m away from the proton)? 27.2V

To be worked at the board.


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Todays agenda:




Electric potential and electric potential energy of asystem of charges.You must be able to calculate both electric potential and electric potential energy for asystem of charged particles (point charges today, charge distributions next lecture).



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Electric Potential Energy of a System of Charges

Electric Potential of a System of Charges

Electric potential energy comes from the interaction betweenpairs of charged particles, so you have to add the potentialenergies ofeach 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 sumof the potential of each charge. (Much easier to calculate!)


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To find the electric potential energy for a system of threecharges, 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

W

e have to add the potential energies ofeach pair of charged particles.


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Electric Potential of a Charge Distribution (details next lecture)

Collection of charges: iPi0 i

q1V .4 r

!TI

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

!TI

Potential at point P.

Well work with this next lecture.


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Example: a 1 QC point charge is located at the origin and a -4QC 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

P

i i 1 2

-6 -6

9

3

q q qV = k = k +

r r r

110 -410= 910 +

3 5

= - 4.210 V

Thanks to Dr. Waddill for the use of these examples.


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Example: how much work is required to bring a +3 QC pointcharge from infinity to point P?

q2q1

3 m

P

4 mx

y

q3

external 3W U q V! ( ! (

external 3 PW q V Vg!

externalW E K U! ( ! ( (

0

0

6 3externalW 3 10 4.2 10! v v

3externalW 1.26 10 J! v

The work done by the external force was negative, so the work done by the electric field waspositive. The electric field pulled q3 in (keep in mind q2 is 4 times as big as q1).

Positive work would have to be done by an external force to remove q3 from P.


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Example: find the total potential energy of the system of threecharges.

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 -69

110 -410 110 310 -410 310U = 9 10 + +

4 3 5

v

-2U = - 2.16 10 Jv


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Todays agenda:







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The Electron Volt

An electron volt (eV) is the energy acquired by a particle ofcharge e when it moves through a potential difference of 1 volt.

U= q V( (

-1

9

1 eV= 1.6 10 C 1 Vv

-191 eV= 1.6 10 Jv

This is a very small amount of energy on a macroscopic scale,but electrons in atoms typically have a few eV (10s to 1000s)of energy.


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Example: on slide 9 we found that the potential energy of thehydrogen atom is about -4.36x10-18joules. 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

v v } v

Hold it! I learned in Chemistry (or high school physics) that theground-state energy of the hydrogen atom is -13.6 eV. Did youmake a physics mistake?

The ground-state energy of the hydrogen atom includes thepositive kinetic energy of the electron, which happens to have amagnitude of half the potential energy. Add KE+PE to getground state energy.


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Remember your Physics 23 hammer equation?

? Af i other i fE E W p !

What goes into Efand Ei? What goes into Wother?

Homework Hints!

Youll need to use starting equations from Physics 23!

? Af i c i fU U W p ! This is also handy:

comment on 23.68 last part, if assigned


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Demo (if not done last time):Professor Tries to Avoid

Debilitating Electrical Shock WhileDemonstrating Van de Graaff Generator

http://en.wikipedia.org/wiki/Van_de_Graaff_generator

Lecture05 Electric Potential; Electric Potential Energy

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