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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
$3%&'• Maybe it would be nice to pick "3#'+ so that ;#'+ = 0.
Electric Potential due to a Point Charge
• We can make ; (#'+ = 0 if we let (#'+ → ∞.
(#'+(
+, 7 ( = 14/01
!($ (
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
!($ (
1/31/2016 24
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 (#'+ → ∞)
1/31/2016 23
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
+
+
+
+
1/31/2016 11
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:
1/31/2016 25
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 �� ��
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