Physics24100Electricity&Op-csLecture6–Chapter23sec.1-3

Fall2017SemesterProfessorKolAck

Work done by a Force

• Definitions from mechanics (Physics 152/172):

• Conservative force: Net work independent of the path.– Total work done: %&'( = *+ − *- (change in potential energy)

.

/ℓ How much work does poor Sisyphus do?

1% = −23 ∙ 1ℓWhat is the change in the potential energy of the rock?

1* = −23 ∙ 1ℓ

Work Done by Electrostatic Force• An electric field exerts a force on a charge.

• If we push it to the left, we do work on the charge:1% = −23 ∙ 1ℓ > 0

– The charge gains potential energy.

• If we release the charge, it accelerates to the right– The charge loses potential energy.– It gains kinetic energy (total energy is conserved).

78 > 9 23/ℓ

! Energy conservation: change in kinetic energy of a particle = - change in potential energy of a particle

! change in kinetic energy of a particle = work done by E-field

Potential Energy vs Electric Potential

• Electric potential difference:

∆; = ? 1;

!= −? 7 ∙ 1ℓ

!= ; − ;!

– This only depends on the electric potential at points < and =…

– Electric potential is a property of the field.

• Difference in potential energy:∆* = :∆;

– This depends on the charge, :…– It is not a property of the field.

SI units: Joules per Coulomb

SI units: Joules

SI Units

• Units of electric potential: Joules per Coulomb

• We will use this so frequently that we define:

1 Volt = 1 Joule/Coulomb• Electric field:

1 Newton/Coulomb = 1 Volt/meter

! Electrostatic force is conservative force

• circulation of electrostatic field is zero

2

NB: there is inductive E-field, for which work over closed path is not 0 (circulation not 0)

ZE · dl = 0

Electric Potential Energy

Lecture 6-8 Electric Potential Energy and Electric Potential

positive charge q0

High U (potential energy)

Low U

negative charge q0

High U

Low U High V (potential)

Low V

Electric field direction

High V

Low V

Electric field direction

0

0

0

( )

( )( ) q

q qU r kr

U r qV r kq r

Properties of the Electric Potential• The electric field points in the direction of decreasing

electric potential.

– As Sisyphus’ rock rolls down hill (direction of 23) it loses potential energy.

• So far, our definition only referred to changes in potential energy and differences in electric potential.– You can add an arbitrary constant to the electric potential

without changing the potential difference.

– But it must be the same value at all points in space.

• We usually define the electric potential as the potential difference relative to a convenient “reference point”.

Calculating Electric Potential• If the electric potential is a property of the field, how

do we calculate it?

• The electric potential of a field at a point "3 is the work per unit charge required to move from the reference point, "3#'+, to the point, "3:

%: = ; "3 − ;#'+ = −? 7 ∙ 1ℓ

$3

$3%&'• Maybe it would be nice to pick "3#'+ so that ;#'+ = 0.

Lecture 6-12 Potential at P due to a point charge q

0

0

( )( ) qU rV r

qqkr

=

=

From ∞

Electric Potential due to a Point Charge

• We can make ; (#'+ = 0 if we let (#'+ → ∞.

(#'+(

+, 7 ( = 14/01

!($ (

1ℓ = −1((

; ( − ; (#'+ = −? 7 ( ∙ 1ℓ#

#%&'

= − 34567

8 9##:

##%&'

= 34567

"# −

"#%&'

Electric Field from a Point Charge

• Electric potential:

; ( = 14/01

!(

• Electric field:

7 = −Z; = −[;[( Z( =

14/01

!($ (

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For a point charge

V(r)

r

to

V(r) versus r for a positive charge at r = 0

kqr

V(r 0)

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Demos:

+ +

R

+

+ + +

+ + +

+

Gauss’ law says the sphere looks like a point charge outside R.

V(r)

R +

+ +

+

+

+

+

+

+

+

V (R) kQR

kQR

Lecture 6-13 Electron Volt

• �V=U/q is measured in volts => 1 V (volt) = 1 J / 1 C

[ ] [ ]

[ ]

J N mV E m VC CN VEC m

⋅= = = ⋅ =

= =

19

1 1 11 | | 1 1.602 10 1J C VeV e V C V−

= ⋅≡ ⋅ ≅ × ⋅

• �V depends on an arbitrary choice of the reference point. • �V is independent of a test charge with which to measure it.

(electron volt)

POTENTIAL DIFFERENCES V2 – V1

ExampleThe walls of this room have an electric potential of zero volts.

This enclosure has an electric potential of -750 kV.

Negatively charged hydrogen ions are produced in the enclosure.

How much kinetic energy do they have when the leave the room?

H-

Lecture 6-1 Electric Potential Energy of a Charge in Electric Field

• Coulomb force is conservative => Work done by the Coulomb force is path independent. • an associate potential energy to charge q0 at any point r in space. ( )U r

It’s energy! A scalar measured in J (Joules)

d l

0dW q E d l Work done by E field

0dU dW q E dl � � Potential energy change of the charge q0

Lecture 6-2 Electric Potential Energy of a Charge (continued)

0dW q E d l

0

( ) ( )r

i

U U r U i

q E dl

' �

�³

0dU dW q E dl � �

i is “the” reference point. Choice of reference point (or point of zero potential energy) is arbitrary.

0

d l

i is often chosen to be infinitely far ( ) f

f

Lecture 6-12 Calculating the Field from the Potential (1)

• We can calculate the electric field from the electric potential starting with

• Which allows us to write

• If we look at the component of the electric field along the direction of ds, we can write the magnitude of the electric field as the partial derivative along the direction s

V �

We,f

q

SVEs

w �

w

dW qE ds

qdV qE ds E ds dV� � �

E from V

xV

Ex

∂= −∂

yVEy

∂= −∂ z

VEz

∂= −∂

Expressed as a vector, E is the negative gradient of V

VE ∇−=!!

We can obtain the electric field E from the potential V by inverting the integral that computes V from E:

∫∫→→

++−=⋅−=→ r

zyx

r

dzEdyEdxEldErV )()(

Lecture 6-16

Electric Field (again)• We can calculate the electric field from the electric

potential:

7 = −Z; = − [;[" \ +

[;[< ^ +

[;[_

a

• This is called the “gradient” of the electric potential.

• Useful relations:[;(()[" = [;

[([(["

( = "$ + <$ + _$ b#b$ =

$#

b#bc =

c#

b#bd =

d#

Z( = (

The “chain rule”…Also works for < and _.

Constant Electric Field

• What electric potential produces a constant electric field, 7 = 7\?

• Electric field:7 = −Z;

– What function will have beb$ = −7, bebc = 0, bebd = 0?

– How about ; "3 = −7"...

• A linear electric potential produces a constantelectric field.

Electric Potential due to several Point Charges

• Electric potential for a single point charge:

; ( = 14/01

!(

• Electric potential at point "3 due to several point charges:

; "3 = 14/01

;!-(--

where (- = "3 − "3- is the distance to each charge.

• ; "3 is a scalar function (no direction).

(when (#'+ → ∞)

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Push q0 “uphill” and its electrical potential energy increases according to

Electrical Potential Energy

U kq0qr

The work required to move q0

initially at rest at is

W kq0qr

.

Work per unit charge is

V kqr

.

Lecture 6-9 Potential Energy of a Multiple-Charge Configuration

(a)

(b)

(c)

1 2 /kq q d

1 31 32 2

2q q qq qk k k

ddq

d++

2 3

1 3 3 41 2 2 4

1 4

2 2

q q q qq q q qk k k kd dq q q qk k

d

d d

d

+

+

+

+

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Example: Electric Potential Energy What is the change in electrical potential energy of a released electron in

the atmosphere when the electrostatic force from the near Earth’s electric field (directed downward) causes the electron to move vertically upwards through a distance d?

1. DU of the electron is related to the work done on it by the electric field:

2. Work done by a constant force on a particle undergoing displacement:

3. Electrostatic Force and Electric Field are related:

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Demo Get energy out

R +

+ +

+

+

+

+

+

+

+ +

r1 r2

charge flow

across fluorescent light bulb Also try an

elongated neon bulb.

DU q DV

V kQR

DV V(r1)V(r2 )

Lecture 6-13 Math Reminder - Partial Derivatives

• Given a function V(x,y,z), the partial derivatives are

• Example: V(x,y,z) = 2xy2 + z3

they act on x, y, and z independently

Meaning: partial derivatives give the slope along the respective direction

wVwx

wVwy

wVwx

2

2

2

4

3

V yxV xyyV zz

w

ww

ww

w

Lecture 6-14 Calculating the Field from the Potential (2)

• We can calculate any component of the electric field by taking the partial derivative of the potential along the direction of that component

• We can write the components of the electric field in terms of partial derivatives of the potential as

• In terms of graphical representations of the electric potential, we can get an approximate value for the electric field by measuring the gradient of the potential perpendicular to an equipotential line

; ; x y zV V VE E Ex y z

w w w � � �

w w w

also written as E V E V �� ��