electric potential energy and voltage

Electricity and Magnetism

Electric Potential Energyand

Voltage

2

Work and Potential EnergyRecall from Mechanics that Emech= K + U is a conserved

quantity for particles that interact via conservative forces and that for changes∆∆∆∆Emech= ∆∆∆∆K + ∆∆∆∆U = 0

The change in potential energy is∆∆∆∆U = Uf ndash Ui = -Wconservative force

If a particle moves a distance ∆∆∆∆r while a constant force F is acting on it then the work done isW = Fmiddot∆∆∆∆r = F ∆∆∆∆r cos(θθθθ) where θθθθ is the angle between the force F and displacement ∆∆∆∆r

There are three special cases θθθθ=00 θθθθ=900 and θθθθ=1800

If the force is not constant the work isf

i

s f

s

s i

W F d s F d s= = sdotint int

3

The Potential Energyin Two Uniform Fields

The gravitational field g near the surface of the Earth is uniform If a particle moves downward from yi to yf the gravitational field will do a positive amount of work

Therefore

grav cos0 ( ) f iW w r mg y y mg y= ∆ deg = minus = ∆

grav gravf iU U U W mg y∆ = minus = minus = minus ∆

Gravitational Potential Energy

4



Therefore




elec cos0 ( ) ( 1)f iW F r qE s s qE s= ∆ deg = minus + = ∆

elec elecf iU U U W qE s∆ = minus = minus = minus ∆

Electric Potential Energy

Similarly for displacements s in a uniform electric field E with s parallel to E

5

Charges in an Electric Field

One difference between a gravity field g and an electric field E is that a mass m interacting with g is always positive while a charge q interacting withE may be either positive or negative

However this is not a problem A positivecharge gains energy as it moves away from the positive plateof a parallel plate capacitor while a negativecharge gains energy as it moves away from the negative plateof the capacitor In either case the charge gainskinetic energy as its potential energy decreases

6

7

Example Conservation of Energy inside a Capacitor

A 20 cm x 20 cm parallel plate capacitor with a 20 mm gap is charged to plusmn10 nC (Later in the year we will see that the electric field between the plates is 283 x 105 NC) First a proton and then an electron are released at the midpoint of the capacitor

(a) What is each particlersquos change in potential energy (∆Uelec) from its release to its collision with a plate

(b) What is each particlersquos kinetic energy as it reaches the plate

8





9

Conceptual Question 1

The electric field of a positively chargedrod (end view shown) causes a negativeparticle to orbit the rod in a closed circularpath as shown

What is the signof the work done on thecharged particle by the electric field of therod

(A) positive (B) zero (C) negative (D) not enough information to tell

10

VoltageIn Chapter 23 we introduced the

concept of an electric field E which can be though of as a normalized force ie E = Fq the field E that would produce a force F on some test charge q

We can similarly define the voltageV as a charge-normalized potential energy ie V=Uelecq the voltage Vthat would give a test charge q an electric potential energy Uelecbecause it is in the field of some other source charges

Just like it is ∆U that really matters and the actual values are arbitrary it is changes in voltage ∆V that we are going to be interested in

We define the unit of voltageas the volt 1 volt = 1 V = 1 JC = 1 NmC

11

What Good isthe Voltage

Like the electric field E the voltage V is an abstract idea It offers an advantage however because it is a scalar quantity while E is a vector yet the two can be converted to each other It is useful because

- The voltage depends only on the charges and their geometries The voltage is the ldquoabilityrdquo of the source charges to have an interaction if a charge q shows up The voltage is present in all space whether or not a charge is there to experience it

- If we know the voltage V throughout a region of space wersquoll immediately know the potential energy U=qV of any charge q that enters that region

12

Example Moving Through a Voltage Difference

A proton (q = 16 x 10-19 C m = 167 x 10-27 kg) with a speed of vi = 2 x105 msenters a region of space where source charges have created a voltage (a) What is the protonrsquos final speed vf after it has moved through a voltage difference of ∆V=100 V (b) What isvf if the proton is replaced by an electron

13

The Voltage Insidea Parallel Plate Capacitor

500 NC to rightE =Consider a parallel-plate capacitor with

Find the voltage difference (potential difference) between the two plates

14

Graphical Representationsof Electric Potential

CC( ) 1

V xV Es d x V

d d

∆ = = minus = ∆ minus

This linear relation can be represented as a graph a set ofequipotential surfaces a contour plot or a 3-D elevation graph

15

Field Lines and Contour LinesField lines and equipotential

contour lines are the most widely used representations to simultaneously show the E field and the electric potential The figure shows the field lines and equipotential contours for a parallel plate capacitor

Remember that both field lines and contours are ldquovirtualrdquo representations not real objects and that their spacing etc is a matter of choice

16

Field Lines and Contour LinesFor a constant electric field if

you know the voltage difference between two points and how far apart the two points are you can calculate the magnitude of the electric field from

VE

x

∆=∆

To get the direction just remember that the voltage decreases as you move in the direction that the electric field points

17

Field Lines and Contour LinesIf the electric field is not constant you can use this method to estimate the strength of the electric field as long as ∆x is small (the smaller ∆x is the closer E is to being constant in that interval)

VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆

We will use this method when we return to this topic and look at the parts that require calculus

18

EField Java Field-Line AppletA special Java applet for plotting electric field lines E-field

gradients and equipotential surfaces of any arrangement of point charges can be found athttpwwwccocaltechedu7Ephys1javaphys1EFieldEFieldhtml

The result looks like this

You must have a Javaapplication available inorder to run this appletYou are encouraged touse it to gain a betterfeeling for electric fields And equipotential lines

19

1 Equipotentials neverintersectother equipotentials (Why)

2 The surface of any staticconductor is an equipotentialsurface The conductor volumeis all at the same potential

3 Field line cross equipotentialsurfaces at right angles (Why)

4 Close equipotentials indicate astrong electric field The voltage V decreases in the direction in which the electric field E points ie energetically ldquodownhillrdquo

5 For any system with a net charge the equipotential surfaces become spheres at large distances

Rules for Equipotentials

20


Which ranking of the voltages at points a-e is correct (Ignore edge effects)

(a) VagtVbgtVcgtVdgtVe

(b) VagtVb=VcgtVd=Ve

(c) Va=VbgtVcgtVd=Ve

(d) Va=Vb=Vc=Vd=Ve

(e) VbgtVagtVcgtVegtVd

1) proton2) electron3) both feel the same force4) neither ndash there is no force5) they feel the same magnitude

force but opposite direction

A proton and an electron are in a constant electric field created by oppositely charged plates You release the proton from the positive side and the electron from the negative side Which feels the larger electric force

E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--


1) proton2) electron3) both feel the same acceleration4) neither ndash there is no acceleration5) they feel the same magnitude

acceleration but opposite direction

A proton and an electron are in a constant electric field created by oppositely charged plates You release the proton from the positive side and the electron from the negative side Which has the larger acceleration

E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--


1) proton2) electron3) both acquire the same KE4) neither ndash there is no change of

KE5) they both acquire the same KE but

with opposite signs

A proton and an electron are in a constant electric field created by oppositely charged plates You release the proton from the positive side and the electron from the negative side When it strikes the opposite plate which one has more KE

E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--


Which requires you to do the most work to move a positive charge from P to points 1 2 3 or 4 All points are the same distance from P


P1

2

3

E

4

1) P rarrrarrrarrrarr 12) P rarrrarrrarrrarr 23) P rarrrarrrarrrarr 34) P rarrrarrrarrrarr 45) all require the same

amount of work

Which requires you to do zero work to move a positive charge from P to points 1 2 3 or 4 All points are the same distance from P


amount of work

P1

2

3

E

4


26

The Voltageof a Point Charge

0

1

4

kq qV

r rπε= = Exampleq = 1 nC r = 1 cm

-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=

The voltage of a point charge (letting the voltage be zero infinitely away from the charges) is given by

We will show that this equation is correct using calculus later in the year For now we are just interested in using it

You would use the given equation to find the voltage at this point due to the source charge q

27


Which ranking of the potential differences is correct

(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28

Visualizing the Voltageof a Point Charge

The potential of a point charge can be represented as a graph a set of equipotential surfaces a contour map or a 3-D elevation graph

Usually it is represented by a graph or a contour map possibly with field lines

+

Which two points are at the samepotential (voltage)

1) A and C2) B and E3) B and D4) C and E5) no pair

A

C

B DEQ


30

The Voltageof Many Charges

i

i i

kqV

r=sum

The principle of superposition allows us to calculate the voltages created by many point charges and then add the up Since the voltage V is a scalar quantity the superposition of potentials is simpler than the superposition of fields

31

Example The Voltageof Two Charges

What is the voltage at point pLet V = 0 at r = infin

p

2

Work and Potential EnergyRecall from Mechanics that Emech= K + U is a conserved

quantity for particles that interact via conservative forces and that for changes∆∆∆∆Emech= ∆∆∆∆K + ∆∆∆∆U = 0

The change in potential energy is∆∆∆∆U = Uf ndash Ui = -Wconservative force

If a particle moves a distance ∆∆∆∆r while a constant force F is acting on it then the work done isW = Fmiddot∆∆∆∆r = F ∆∆∆∆r cos(θθθθ) where θθθθ is the angle between the force F and displacement ∆∆∆∆r

There are three special cases θθθθ=00 θθθθ=900 and θθθθ=1800

If the force is not constant the work isf

i

s f

s

s i

W F d s F d s= = sdotint int

3



Therefore




4



Therefore








5




6

7





8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

3



Therefore




4



Therefore








5




6

7





8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

4



Therefore








5




6

7





8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

5




6

7





8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

6

7





8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

7





8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

8





9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

9





10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

10






11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

11





12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

12



13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

13




14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

14


CC( ) 1

V xV Es d x V

d d



15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

15




16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

16



VE

x

∆=∆


17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

17


VE

x

∆asymp∆

or more exactly

0lim

x

V dVE

x dx∆ rarr

∆= =∆


18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

18





19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

19







20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

20






(d) Va=Vb=Vc=Vd=Ve





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p




E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--





E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p




E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p



with opposite signs


E

electronelectron

protonproton

E

ElectronElectron

ProtonProton++

--




P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p



P1

2

3

E

4


amount of work



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p



amount of work

P1

2

3

E

4


26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

26


0

1

4

kq qV


-99 2 2

-2

(10 10 C)(90 10 Nm C )

(10 10 m)

900 V

kqV

r=

times= timestimes

=




27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

27



(a) ∆V12gt ∆ V23gt ∆ V13

(b) ∆ V12lt ∆ V23lt ∆ V31

(c) ∆ V12lt ∆ V23= ∆ V13

(d) ∆ V12= ∆ V23gt ∆ V13

(e)∆ V12= ∆ V23= ∆ V13

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

28




+



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p



A

C

B DEQ


30


i

i i

kqV

r=sum


31



p

30


i

i i

kqV

r=sum


31



p

31



p

electric potential energy and voltage

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