Download - From last time(s)…
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Oct. 4, 2007 1
From last time(s)…
Work, energy, and (electric) potential Electric potential and charge Electric potential and electric field.
Electric charges, forces, and fields Motion of charged particles in fields.
Today…
No honors lecture this week
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Oct. 4, 2007 2
Forces, work, and energy Particle of mass m at rest Apply force to particle - what happens? Particle accelerates
Stop pushing - what happens? Particle moves at constant speed Particle has kinetic energy
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Oct. 4, 2007 3
Work and energy Work-energy theorem:
Change in kinetic energy of isolated particle = work done
€
dW =r F ⋅d
r s = Fdscosθ
Total work
€
ΔK = dWstart
end
∫ =r F • d
r s
start
end
∫
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Oct. 4, 2007 4
Electric forces, work, and energy
Consider bringing two positive charges together They repel each other
Pushing them together requires work Stop after some distance How much work was done?
+ +
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Oct. 4, 2007 5
Calculating the work E.g. Keep Q2 fixed, push Q1 at constant velocity
Net force on Q1 ?
Force from hand on Q1 ?
+ +Q1 Q2
Zero
€
1
4πεo
Q1Q2
R2
Total work done by hand
€
rF • d
r s
start
end
∫ = ke
Q1Q2
R − x( )2 dx
xinitial
x final
∫
= ke
Q1Q2
R − x xinitial
x finalForce in direction of motion
R
+
xinitialxfinal
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Oct. 4, 2007 6
Conservation of Energy
Work done by hand
€
=ke
Q1Q2
R − x xinitial
x final
= ke
Q1Q2
R − x final
− ke
Q1Q2
R − x initial
= ke
Q1Q2
rfinal
− ke
Q1Q2
rinitial
> 0
Where did this energy go?
Energy is stored in the electric field as electric potential energy
for pos charges
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Oct. 4, 2007 7
Electric potential energy of two charges
Define electric potential energy U so that
€
Wexternal = ΔK + ΔUWork done on system
Change in kinetic energy
Change in electric potential energy
Works for a two-charge system if
€
U r( ) = ke
Q1Q2
r+ const
Define: potential energy at infinite separation = 0
€
U r( ) = ke
Q1Q2
rThen
Units of Joulesfor two charges
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Oct. 4, 2007 8
Quick QuizTwo balls of equal mass and equal charge are held
fixed a distance R apart, then suddenly released. They fly away from each other, each ending up moving at some constant speed.
If the initial distance between them is reduced by a factor of four, their final speeds are
A. Two times bigger
B. Four times bigger
C. Two times smaller
D. Four times smaller
E. None of the above
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Oct. 4, 2007 9
More About U of 2 Charges
Like charges U > 0 and work must be done to bring the charges together since they repel (W>0)
Unlike charges U < 0 and work is done to keep the charges apart since the attract one the other (W<0)
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Oct. 4, 2007 10
Electric Potential Energy of single charge
Work done to move single charge near charge distribution. Other charges provide the force, q is charge of interest.
+++
+q
€
rF =
r F q1
+r F q2
+r F q3
q1
q2
q3
€
U =r F • d
r s ∫ =
r F q1
+r F q 2 +
r F q3( ) • d
r s ∫
= Uq1r1( ) + Uq 2 r2( ) + Uq 3 r3( )
= kq1q
r+ k
q2q
r+ k
q3q
rSuperposition of individual interactionsGeneralize to continuous charge distribution.
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Oct. 4, 2007 11
Electric potential
Electric potential V usually created by some charge distribution.
V used to determine electric potential energy U of some other charge q
V has units of Joules / Coulomb = Volts
€
ΔU =r F Coulomb • d
r s ∫ = q
r E • d
r s ∫ = q
r E • d
r s ∫
Electric potential ΔU energy proportional to charge q
€
ΔU /q ≡ V = Electric potential
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Oct. 4, 2007 12
Electric potential of point charge
Consider one charge as ‘creating’ electric potential, the other charge as ‘experiencing’ it
Q
q
€
Vq r( ) =UQq r( )
q= ke
Q
r
€
UQq r( ) = ke
r
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Oct. 4, 2007 13
Electric Potential of point charge
Potential from a point charge Every point in space has a
numerical value for the electric potential
y
x
€
V =kQ
r
Distance from ‘source’ charge +Q
+Q
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Oct. 4, 2007 14
Potential energy, forces, work
U=qoV Point B has greater potential
energy than point A Means that work must be done
to move the test charge qo
from A to B. This is exactly the work to
overcome the Coulomb repulsive force.
Ele
ctric
pot
entia
l ene
rgy=
qoV
A
B
qo > 0Work done = qoVB-qoVA =
€
− r
F Coulomb( ) • dr l
A
B
∫
Differential form:
€
qodV = −r F Coulomb • d
r l
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Oct. 4, 2007 15
Quick QuizTwo points in space A and B have electric potential
VA=20 volts and VB=100 volts. How much work does it take to move a +100µC charge from A to B?
A. +2 mJ
B. -20 mJ
C. +8 mJ
D. +100 mJ
E. -100 mJ
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Oct. 4, 2007 16
V(r) from multiple charges
Work done to move single charge near charge distribution. Other charges provide the force, q is charge of interest.
q
q1
q2
q3
€
U =r F • d
r s ∫ =
r F q1
+r F q 2 +
r F q3( ) • d
r s ∫
= Uq1r1( ) + Uq 2 r2( ) + Uq 3 r3( )
= kq1q
r+ k
q2q
r+ k
q3q
r
= q kq1
r+ k
q2
r+ k
q3
r
⎛
⎝ ⎜
⎞
⎠ ⎟
= q Vq1 r( ) + Vq2 r( ) + Vq 3 r( )( )Superposition of individual electric potentials
€
V r( ) = Vq1 r( ) + Vq 2 r( ) + Vq 3 r( )
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Oct. 4, 2007 17
Quick Quiz 1 At what point is the electric potential zero for this
electric dipole?
+Q -Q
x=+ax=-aA
B
A. A
B. B
C. Both A and B
D. Neither of them
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Oct. 4, 2007 18
Superposition: the dipole electric potential
+Q -Q
x=+ax=-aSuperposition of• potential from +Q• potential from -Q
Superposition of• potential from +Q• potential from -Q
+ =
V in plane
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Oct. 4, 2007 19
Electric Potential and Field for a Continuous Charge Distribution
If symmetries do not allow an immediate application of the Gauss’ law to determine E often it is better to start from V!
Consider a small charge element dq The potential at some point due to this charge element is
To find the total potential, need to integrate over all the elements
This value for V uses the reference of V = 0 when P is infinitely far away from the charge distribution
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Oct. 4, 2007 20
Quick QuizTwo points in space have electric potential VA=200V & VB=150V.
A particle of mass 0.01kg and charge 10-4C starts at point A with zero speed. A short time later it is at point B.
How fast is it moving?
A. 0.5 m/s
B. 5 m/s
C. 10 m/s
D. 1 m/s
E. 0.1 m/s
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Oct. 4, 2007 21
E-field and electric potential If E-field known,
don’t need to know about charges creating it. E-field gives force From force, find work to move charge q
+++
+
€
rE
q
€
rF elec = q
r E
€
ΔU =r F me • d
r s ∫ = −q
r E • d
r s ∫ = −q
r E • d
r s ∫
€
ΔV = ΔU /q = −r E ⋅d
r s ∫Electric potential
Non-constant potential Non-zero E-field
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Oct. 4, 2007 22
Potential of spherical conductor
Zero electric field in metal -> metal has constant potential Charge resides on surface,
so this is like the spherical charge shell. Found E = keQ / R2 in the radial direction. What is the electric potential of the conductor?
€
V R( ) −V ∞( ) = − E∞
R
∫ • ds
= ER
∞
∫ • ds
Integral along some path, from point on surface to inf.
difficult path
easy path
Easy because is same direction as E,
€
E • ds = E dr = Edr
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Oct. 4, 2007 23
Electric potential of sphere
Conducting spheres connected by conducting wire.Same potential everywhere.
€
V R( ) −V ∞( ) = − E∞
R
∫ dr = kQ
r2R
∞
∫ dr
= −kQ
r R
∞
= kQ
R
So conducting sphere of radius R carrying charge Q is at a potential
€
kQ /R
R1R2
Q1 Q2
But not same everywhere
But not same everywhere
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Oct. 4, 2007 24
Connected spheres
Since both must be at the same potential,
€
kQ1
R1
=kQ2
R2
⇒Q1
Q2
=R1
R2
Surface charge densities?
€
σ =Q
4πR2⇒
σ 1
σ 2
=R2
R1
Charge proportional to radius
Charge proportional to radius
Surface charge density proportional to 1/R
Surface charge density proportional to 1/R
Electric field?
Since
€
E =σ
2εo
Local E-field proportional to 1/R (1/radius of curvature)
Local E-field proportional to 1/R (1/radius of curvature)
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Oct. 4, 2007 25
Varying E-fields on conductor
Expect larger electric fields near the small end. Can predict electric field proportional to local radius of curvature.
Large electric fields at sharp points, just like square Fields can be so strong that air is ionized and ions accelerated.
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Oct. 4, 2007 26
Quick Quiz
Four electrons are added to a long wire. Which of the following will be the charge distribution?
A)
B)
C)
D)
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Oct. 4, 2007 27
Conductors: other geometries Rectangular conductor (40
electrons) Edges are four lines Charge concentrates at
corners Equipotential lines closest
together at corners So potential changes faster
near corners. So electric field is larger at
corners.
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Oct. 4, 2007 28
E-field and potential energy
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Oct. 4, 2007 29
Zero
What is electric potential energy of isolated charge?
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Oct. 4, 2007 30
The Electric Field
is the Electric Field It is independent of the test charge,
just like the electric potential It is a vector, with a magnitude and direction, When potential arises from other charges,
= Coulomb force per unit charge on a test charge due to interaction with the other charges.
€
qodV = −r F Coulomb • d
r l ⇒
dV =−
r F Coulomb
qo
⎛
⎝ ⎜
⎞
⎠ ⎟• d
r l
€
=− r
E • dr l
€
rE
€
rE
We’ll see later that E-fields in electromagnetic waves exist w/o charges!
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Oct. 4, 2007 31
Electric field and potential
Electric field strength/direction shows how the potential changes in different directions
For example, Potential decreases in direction of local E field at rate Potential increases in direction opposite to local E-field at rate potential constant in direction perpendicular to local E-field
€
dV = −r E • d
r l Said before that
€
rE • d
r l = 0( )
€
∝ r
E
€
∝ r
E
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Oct. 4, 2007 32
Potential from electric field
Electric field can be used to find changes in potential
Potential changes largest in direction of E-field.
Smallest (zero) perpendicular to E-field
€
dV = −r E • d
r l
€
dV = −r E • d
r l
€
dr l
€
rE
V=Vo
€
V = Vo −r E d
r l
€
V = Vo +r E d
r l
€
dr l
€
dr l
€
V = Vo
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Oct. 4, 2007 33
Quick Quiz 3
Suppose the electric potential is constant everywhere. What is the electric field?
A) Positive
B) Negative
C) Zero
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Oct. 4, 2007 34
Electric Potential - Uniform Field
Constant E-field corresponds to linearly increasing electric potential
The particle gains kinetic energy equal to the potential energy lost by the charge-field system
€
ΔV = −r E • d
r s ⇒ VB −VA = −
r E
A
B
∫ • dr s
€
=− r
E A
B
∫ dr s = −E d
r s
A
B
∫ = −Ex
€
rE || d
r s
€
rE || d
r s E cnstE cnst
A B
x
€
E = −ΔV
d= −
VB −VA
d
+
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Oct. 4, 2007 35
Electric field from potential
Said before that Spell out the vectors:
This works for
€
dV = −r E • d
r l
€
dV = − Exdx + Eydy + E zdz( )
€
Ex = −dV
dx, Ey = −
dV
dy, E z = −
dV
dz
Usually written
€
r E = −
r ∇V = −
dV
dx,dV
dy,dV
dz
⎛
⎝ ⎜
⎞
⎠ ⎟
€
r E = −
r ∇V = −
dV
dx,dV
dy,dV
dz
⎛
⎝ ⎜
⎞
⎠ ⎟
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Oct. 4, 2007 36
Equipotential lines
Lines of constant potential In 3D, surfaces of constant potential
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Oct. 4, 2007 37
Electric Field and equipotential lines for + and - point charges
The E lines are directed away from the source charge
A positive test charge would be repelled away from the positive source charge
The E lines are directed toward the source charge
A positive test charge would be attracted toward the negative source charge
Blue dashed lines are equipotential
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Oct. 4, 2007 38
Quick Quiz 1
C
C
C m
m
m
1. W = +19.8 mJ2. W = 0 mJ3. W = -19.8 mJ
Question: How much work would it take YOU to assemble 3 negative charges?
Likes repel, so YOU will still do positive work!
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Oct. 4, 2007 39
Work done to assemble 3 charges
W1 = 0
1C
3C
2C m
m
m
• W2 = k q1 q2 /r
• W3 = k q1 q3/r + k q2 q3/r (9109)(110-6)(310-6)/5 + (9109)(210-6)(310-6)/5
=16.2 mJ
• W = +19.8 mJ• WE = -19.8 mJ• UE = +19.8 mJ
=(9 109)(1 10-6)(2 10-6)/5 =3.6 mJ
q3
q2q1
Similarly if they are all positive:
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Oct. 4, 2007 40
Quick Quiz 2
Q
Q
Q m
m
m
1. positive2. zero3. negative
The total work required for YOU to assemble the set of charges as shown below is:
€
W1 = 0
W2 = kQ(−Q)
d
W3 = kQQ
d+ k
Q(−Q)
d
Total work = −kQ2
d
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Oct. 4, 2007 41
Why ΔU/qo ?
Why is this a good thing?
ΔV=ΔU/qo is independent of the test charge qo
Only depends on the other charges. ΔV arises directly from these other charges, as described last time.
Last week’s example: electric dipole potential
-Q +Q
x=+ax=-a
Superposition of• potential from +Q• potential from -Q
Superposition of• potential from +Q• potential from -Q
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Oct. 4, 2007 42
Dipole electric fields
Since most things are neutral, charge separation leads naturally to dipoles.
Can superpose electric fields from charges just as with potential
But E-field is a vector, -add vector components
+Q -Q
x=+ax=-a
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Oct. 4, 2007 43
Quick Quiz 2
In this electric dipole, what is the direction of the electric field at point A?
A) Up
B) Left
C) Right
D) Zero +Q -Q
x=+ax=-a
A
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Oct. 4, 2007 44
Dipole electric fields
+Q -Q
Note properties of E-field lines
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Oct. 4, 2007 45
Conservative forces
Conservative Forces: the work done by the force is independent on the path and depends only on the starting and ending locations.
It is possible to define the potential energy U Wconservative = Δ U = Uinitial - Ufinal = = -
(Kfinal - Kinitial) = -ΔK
Fg
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Oct. 4, 2007 46
Potential Energy of 2 charges Consider 2 positive charged particles. The electric force between them is
The work that an external agent should do to bring q2 at a distance rf from q1 starting from a very far away distance is equal and opposite to the work done by the electric force. Charges repel W>0!
r12
F
€
W = F ⋅dr∞
r12∫ = −WE
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Oct. 4, 2007 47
Potential Energy of 2 charges
Since the 2 charges repel, the force on q2 due to q1
F12 is opposite to the direction of motion The external agent F = -F12 must do positive work!
W > 0 and the work of the electric force WE < 0
r12
Fdr
€
W = −keq1q2 −1
r
⎡ ⎣ ⎢
⎤ ⎦ ⎥
∞
r12
= ke
q1q2
r12
€
W = −WE = − F ⋅dr∞
r12
∫ = − Fdr∞
r12∫ = − ke
q1q2
r2dr
∞
r12∫
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Potential Energy of 2 charges
Since WE = - U = Uinitial - Ufinal = = -W W = U
We set Uinitial = U( ) = 0 since at infinite distance the force becomes null
The potential energy of the system is
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More than two charges?
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U with Multiple Charges
If there are more than two charges, then find U for each pair of charges and add them
For three charges: