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1. Add up the contribution of all point charges at this point
V A =1
4p 0qir ii
q1 r 1
q2 r 2
A
2. Travel along a path from point very far away to the locationof interest and add up at each step: E d
l
V A = E dl
A
q1
q2
A E
dl
Potential at a Certain Location
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A
Example: E = 0 inside a charged metal sphere, but V is not!
Common PitfallAssume that the potential V at a location is defined by theelectric field at this location. E
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A negat ive test charge Q = -0.6C was moved from point A topoint B In a uniform electric field E=5N/C . The test charge is at res t before and after the move. The distance between A andB is 0.5m and the line connecting A and B is perpendicular tothe electric field. How much work was done by the net externalforce while moving the test charge from A to B?
A
B
0.5mE = 5 N/C
A. 1.5JB. 0JC. 1.5JD. 3.0JE. 3.0J
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After moving the -0.6C test charge from A to B, it was thenmoved from B to C along the electric field line. The test chargeis at rest before and after the move. The distance between Band C also is 0.5m. How much work was done by the netexternal force while moving the test charge from A to C?
A. 1.5JB. 0J
C. 1.5JD. 3.0JE. -3.0J
A
CB
0.5m E = 5 N/C
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+1.5 J of work was done by the net external force while movingthe -0.6 C test charge from B to C. The test charge is at rest before and after the move. What is the voltage differencebetween B and C, and at which point is the voltage larger?
A. 2.5 V, voltage higher at B.B. 2.5 V, voltage higher at C.C. 0.9 V, voltage higher at B.D. 1.5 V, voltage higher at B.E. 1.5 V, voltage higher at C.
A
E = 5 N/C
B
0.5m
C0.5m
D
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A compass needle turns and points in a particular direction
there is something which interacts with it
Magnetic field ( B): whatever it is that is detected by a compass
Compass: similar to electric dipole
Magnetic Field
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Magnetic fields are produced by moving charges
Current in a wire: convenient source of magnetic field
Static equilibrium: net motion of electrons is zero
Can make electric circuit with continuous motion of electrons
The electron current ( i) is the number of electrons per secondthat enter a section of a conductor.
Counting electrons: complicated
Indirect methods:measure magnetic fieldmeasure heating effect
Electron Current
Both are proportional to the electron current
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Imagine an electric circuit:
What is the effect on the compass needle?
What if we switch polarity?
What if we run wire under compass?
What if we change the current or there is no current in the wire?
The Magnetic Effects of Currents
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Experimental results: The magnitude of B depends on the amount of current A wire with no current produces no B B is perpendicular to the direction of current B under the wire is opposite to B over the wire
Oersted effect: discovered in 1820 by H. Ch. rsted
How does the field around a wire look?
The Magnetic Effects of Currents
Hans Christian rsted(1777 - 1851)
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Principle of superposition: wire Earthnet B B B +=
What can you say about the magnitudes of B Earth and Bwire ?
What if B Earth were much larger than Bwire ?
The Magnetic Effects of Currents
The moving electrons in a wire create a magnetic field
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A current-carrying wire is oriented N-S and laid on top of acompass. The compass needle points 27 o west. What is themagnitude and direction of the magnetic field created by thewire Bwire if the magnetic field of Earth is B Earth = 2
10 -5 T(tesla).
wire Earthnet B B B +=
qtan Earthwire B B =
27tan1025
=
Twire BT5101 wire B
Exercise
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Nikola Tesla (1856-1943)High tension coil demonstration
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A B = A x A y A z
B x B y B z
=
A y B z A z B y A z B x A x B z A x B y A y B x
=
i
j
k
A x A y A z B x B y B z
= A y B z A z B y( )
i A x B z A z B x( )
j + A x B y A y B x( )
k
Calculate magnitude:
A B = A Bsin q
Right-hand rule
The Cross Product
Calculate direction:
2
0
4 r
r vq B
=
p
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Question
W h a t i s t he d ir e ct i o n o f
< 0 , 0 , 3 > x < 0 , 4 , 0 > ?
A ) +x B ) x C ) +y D ) y E ) z er o m ag n it u d e
A
B A =< 0,0,3 > ; B =< 0,4,0 >
0 0 30 4 0
= (0 12)
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What is B straight ahead?
2
0
4 r
r vq B
=
p
What if the charge is negative?
Exercise
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2
0
4 r r vq
B
=p
v
r B1
B2
B3
Which is larger, B1 or B3 ?
Which is larger, B1 or B2 ?
Distance Dependence
B =
0
4pqvr 2
sin q
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2
0
4 r r vq
B
= p
v
r B1
v
r B
v
r B1
Magnetic field depends on qv: Positive and negative charges producethe same B if moving in oppositedirections at the same speed
For the purpose of predicting B wecan describe current flow in terms of
conventional current
positive
moving charges.
Moving Charge Sign Dependence
B =
0
4pqvr 2
sin q+
-
-
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QuestionAn electron passing through the origin is traveling at a constantvelocity in the negative y direction. What is the direction of themagnetic field at a point on the posi ti ve z axis?
A) -x
B) +xC) -zD) +zE) No magnetic field
x
z
y
v
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What would the direction of conventional current have to be?
A current-carrying wire lies ontop of a compass. What is thedirection of the electron currentin this wire?
Exercise
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Electric fields: produced by charges
Magnetic fields: produced by moving charges
charged tape
Any magnetic field?0
4 20 =
=
r r vq B p
0
4 20
=
r
r vq
B p
Frame of Reference
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20
4 r r vq B
= p
Must use the velocities of the charges as you observe them in
your reference frame !
There is a deep connection between electric field and magnetic
fields (Einstein
s special theory of relativity)
Frame of Reference
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If we suddenly change the current in a wire:Magnetic field will not change instantaneously.
Electron and positron collide:Produce both electric and magnetic field, these fields existeven after annihilation.
Changes propagate at speed of light
2
0
4 r r vq
B
=p
There is no time in Biot-Savart law: Speed of moving charges must be small
Retardation
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A steady flow of charges in one direction will create amagnetic field. How can we cause charges to flow steadily?
Need to find a way to produce and sustain E in a wire.
Use battery
Electron Current
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nelectrons
m 3
A m
2
( ) v m
s
t s( ) = nAv t electrons
Electron current: i =# electrons
s= nAv
mobileelectron
density
wireCross sectional
area
Averagedrift
speed
Electron Current
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Typical electron current in a circuit is ~ 10 18 electrons/s.What is the drift speed of an electron in a 1 mm thick copperwire of circular cross section?
328104.8 mnvnA=s
electrons#
A =p D2
4
3.14 1 10 3 m( )2
4= 8 10 7 m 2
v =10 18 s -1
nA
10 18 s -1
8.4 10 28 m 3( )8 10 7 m 2( ) = 1.5 10 5 m/s
Typical Mobile Electron Drift Speed
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m/s5105.1 =vHow much time would it take for a particular electron to movethrough a piece of wire 30 cm long?
hours!sm/s
m5.5102
105.1
30 45
=
==
.v s
t
How can a lamp light up as soon as you turn it on?
Typical Mobile Electron Drift Speed
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Observing magnetic field around copper wire: Can we tell whether the current consists of electrons or positive
holes
?
In some materials current moving charges are positive: Ionic solution
Holes
in some materials (same charge as electron but +)
2
0
4 r
r vq B
=
p
2
0
4 r
r ve B
=
p
=
0
4p
e( ) v( ) r
r 2
Conventional Current
The prediction of the Biot-Savart law is exactly the same in either case.
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2
0
4 r
r ve B
=
p
=
0
4p
e( ) v( ) r
r 2
Metals: current consists of electronsSemiconductors:
n-type electrons
p-type positive holes Most effects are insensitive to the sign of mobile charges:introduce conventional current :
vnAqiq I ==
Conventional Current
Units: C/s A (Ampere)
Andr Marie Ampre(1775 - 1836)
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A typical electron current in a circuit is 10 18 electrons/s.What is the conventional current?
I = q i = 1.6 10 19 C( )10 18 s -1( ) = 0.16 A
Exercise
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Superposition principle is valid
Bi = 04
qivi
r ir i
2
B = Bii
= 04
qivi
r ir
i
2i
=
ir r vq
B 1
4 20
l nAr
r vq B
=
2
0
4
vnAqiq I ==
20
4 r r l I
B
=
The Biot-Savart law for ashort length of thin wire
The Biot-Savart Law for Currents
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