Transcript Potential at a Certain Location

Potential at a Certain Location
1. Add up the contribution of all point charges at this point
1 qi
VA = å
i 4pe 0 ri
q1
q2
r2
r1 A
2. Travel along a path from point very far away to the location
of interest and add up -E · dl at each step:
q2
A
dl
E
VA = - ò E · dl
¥
q1
A
Common Pitfall
Assume that the potential V at a location is defined by the
electric field E at this location.
A
Example: E = 0 inside a charged metal sphere, but V is not!
A negative test charge Q = -0.6C was moved from point A to
point B In a uniform electric field E=5N/C. The test charge is at
rest before and after the move. The distance between A and
B is 0.5m and the line connecting A and B is perpendicular to
the electric field. How much work was done by the net external
force while moving the test charge from A to B?
A
0.5m
B
A.
B.
C.
D.
E.
1.5J
0J
–1.5J
3.0J
–3.0J
E = 5 N/C
After moving the -0.6C test charge from A to B, it was then
moved from B to C along the electric field line. The test charge
is at rest before and after the move. The distance between B
and C also is 0.5m. How much work was done by the net
external force while moving the test charge from A to C?
A
E = 5 N/C
0.5m
A.
B.
C.
D.
E.
1.5J
0J
–1.5J
3.0J
-3.0J
B
C
Instead of moving the test charge from A to B then to C, it is
moved from A to D and then back to C. The test charge is at
rest before and after the move. How much work was done by
the net external force while moving the test charge this time?
A
D
0.5m
B
A.
B.
C.
D.
E.
0.5m
C
1.5J
0J
–1.5J
Infinitely big
Do not know at this time.
E = 5 N/C
+1.5 J of work was done by the net external force while moving
the -0.6 C test charge from B to C. The test charge is at rest
before and after the move. What is the voltage difference
between B and C, and at which point is the voltage larger?
A
D
0.5m
B
A.
B.
C.
D.
E.
0.5m
C
2.5 V, voltage higher at B.
2.5 V, voltage higher at C.
0.9 V, voltage higher at B.
1.5 V, voltage higher at B.
1.5 V, voltage higher at C.
E = 5 N/C
Chapter 18
Magnetic Field
Magnetic Field
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
Electron Current
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 second
that enter a section of a conductor.
Counting electrons: complicated
Indirect methods:
measure magnetic field
measure heating effect
Both are proportional to the electron current
Detecting Magnetic Fields
We use a magnetic compass as a detector of B.
How can we be sure that it does not simply respond to
electric fields?
Compass needle:
Interacts with iron, steel – even if they are neutral
Unaffected by aluminum, plastic etc., though
charged objects polarize and interact with these
materials
Points toward North pole – electric dipole does not do that
The Magnetic Effects of Currents
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
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. Ørsted
How does the field around a wire look?
Hans Christian Ørsted
(1777 - 1851)
The Magnetic Effects of Currents
The moving electrons in a wire create a magnetic field
Principle of superposition: Bnet = BEarth + Bwire
What can you say about the magnitudes of BEarth and Bwire?
What if BEarth were much larger than Bwire?
Exercise
A current-carrying wire is oriented N-S and laid on top of a
compass. The compass needle points 27o west. What is the
magnitude and direction of the magnetic field created by the
wire Bwire if the magnetic field of Earth is BEarth= 2 10-5 T
(tesla).
Bnet = BEarth + Bwire
Bwire = BEarth tan q
Bwire = 2 ´ 10-5 T ´ tan 27
Bwire » 1 ´ 10-5 T
Biot-Savart Law for a Single Charge
q
Electric field of a point charge: E =
rˆ
2
4pe 0 r
1
Moving charge makes a curly magnetic field:
B units: T (tesla) = kg s-2A-1
m0 qv ´ rˆ
B=
4p r 2
2
m0
-7 T × m
= 10
4p
C × m/s
Jean-Baptiste Biot
(1774-1862)
Felix Savart
(1791-1841)
Nikola Tesla
(1856-1943)
Nikola Tesla (1856-1943)
High tension coil demonstration
The Cross Product
æ A ö æ B ö æ A B -A B
z y
ç x ÷ ç x ÷ ç y z
A ´ B = ç A y ÷ ´ ç By ÷ = ç Az Bx - Ax Bz
çç
÷÷ çç
÷÷ ç
ç
è Az ø è Bz ø è Ax By - Ay Bx
= ( A B - A B ) iˆ - ( A B - A B ) ĵ + ( A B
y
z
z
y
x
z
z
x
x
y
ö
ˆ
i
÷
÷ = Ax
÷
÷
Bx
ø
- Ay Bx ) k̂
Calculate magnitude:
A ´ B = A B sinq
Calculate direction:
Right-hand rule
m0 qv ´ rˆ
B=
4p r 2
ĵ
k̂
Ay
Az
By
Bz
Question
A´ B
A =< 0,0, 3 >; B =< 0, 4,0 >
What is the direction of
< 0, 0, 3> x < 0, 4, 0>?
𝑖 𝑗
0 0
0 4
𝑘
3 = (0 − 12)𝑖
0
A) +x
B) –x
C) +y
D) –y
E) zero magnitude
Two-dimensional Projections
 a vector (arrow) is facing into the screen
 a vector (arrow) is facing out of the screen
m0 qv ´ rˆ
B=
4p r 2
B
B
B
r
v
B
B
Why must the field change direction above and below the dashed line?
Exercise
m0 qv ´ rˆ
B=
4p r 2
What is B straight ahead?
What if the charge is negative?
Distance Dependence
B2
m0 qv ´ rˆ
B=
4p r 2
B1
B3
r
v
m0 qv
B=
sinq
2
4p r
Which is larger, B1 or B3 ?
Which is larger, B1 or B2 ?
Moving Charge Sign Dependence
m0 qv ´ rˆ
B=
4p r 2
B1
r
m0 qv
B=
sinq
2
4p r
Magnetic field depends on qv:
Positive and negative charges produce
the same B if moving in opposite
directions at the same speed
For the purpose of predicting B we
can describe current flow in terms of
‘conventional current’ – positive
moving charges.
v
+
B
r
-
v
B1
r
v
-
Question
An electron passing through the origin is traveling at a constant
velocity in the negative y direction. What is the direction of the
magnetic field at a point on the positive z axis?
y
A)
B)
C)
D)
E)
-x
+x
-z
+z
No magnetic field
x
v
z
Exercise
A current-carrying wire lies on
top of a compass. What is the
direction of the electron current
in this wire?
What would the direction of conventional current have to be?
𝐵
𝑒− 𝑒− 𝑒− 𝑒−
Frame of Reference
Electric fields: produced by charges
Magnetic fields: produced by moving charges
B=
m0 qv ´ rˆ
=0
2
4p r
Any magnetic field?
m0 qv ´ rˆ
B=
¹0
2
4p r
charged tape
Frame of Reference
m0 qv ´ rˆ
B=
2
4p r
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)
Retardation
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 exist
even after annihilation.
Changes propagate at speed of light
m0 qv ´ rˆ
B=
4p r 2
There is no time in Biot-Savart law:
Speed of moving charges must be small
Electron Current
A steady flow of charges in one direction will create a
magnetic 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
mö
æ electrons ö
2 æ
çè n
÷ø A m çè v ÷ø ( Dt s ) = nAv Dt electrons
3
m
s
(
mobile
electron
density
)
wire
Cross sectional
area
Average
drift
speed
# electrons
Electron current: i =
= nAv
s
Typical Mobile Electron Drift Speed
Typical electron current in a circuit is ~ 1018 electrons/s.
What is the drift speed of an electron in a 1 mm thick copper
wire of circular cross section?
# electrons
= nAv
s
n » 8.4 ´ 1028 m-3
3.14 × (1 ´ 10 m )
pD
A=
»
= 8 ´ 10 -7 m 2
4
4
-3
2
2
1018 s-1
1018 s-1
-5
v=
»
=
1.5
´
10
m/s
28
-3
-7
2
nA
8.4 ´ 10 m 8 ´ 10 m
(
)(
)
Typical Mobile Electron Drift Speed
-5
v = 1.5 ´ 10 m/s
How much time would it take for a particular electron to move
through a piece of wire 30 cm long?
s
0.3 m
4
t= =
=
2
´
10
s » 5.5 hours!
-5
v 1.5 ´ 10 m/s
How can a lamp light up as soon as you turn it on?
Conventional Current
In some materials current moving charges are positive:
Ionic solution
“Holes” in some materials (same charge as electron but +)
Observing magnetic field around copper wire:
Can we tell whether the current consists of electrons or positive
‘holes’?
m0 qv ´ rˆ
B=
4p r 2
m0 ev ´ rˆ m0 ( -e) ( -v ) ´ r̂
B=
=
2
4p r
4p
r2
The prediction of the Biot-Savart law is exactly the same in either case.
Conventional Current
m0 ev ´ rˆ m0 ( -e) ( -v ) ´ r̂
B=
=
2
4p r
4p
r2
Metals: current consists of electrons
Semiconductors:
n-type – electrons
p-type – positive holes
Most effects are insensitive to the sign of mobile charges:
introduce conventional current:
I = q i = q nAv
Units: C/s  A (Ampere)
André Marie Ampère
(1775 - 1836)
Exercise
A typical electron current in a circuit is 1018 electrons/s.
What is the conventional current?
(
)(
)
I = q i = 1.6 ´ 10 -19 C 1018 s-1 = 0.16 A
The Biot-Savart Law for Currents
Superposition principle is valid
m0 qi vi ´ r̂i
DBi =
4p ri2
I = q i = q nAv
The Biot-Savart law for a
short length of thin wire
m0 qi vi ´ r̂i
DB = å DBi = å
2
4
p
r
i
i
i
m0 qv ´ rˆ
DB =
1
å
2
4p r
i
m0 qv ´ rˆ
DB =
nADl
2
4p r
m0 IDl ´ rˆ
DB =
4p r 2
Biot-Savart Law
Moving charge produces a curly magnetic field
m0 qv ´ rˆ
B=
Single Charge:
4p r 2
Current:
m0 IDl ´ rˆ The Biot-Savart law for a
DB =
short length of thin wire
4p r 2
I = q i = q nAv
B units: T (Tesla) = kg s-2A-1
2
m0
T
×
m
= 10-7
4p
C × m/s