Download Magnetic Fields

W06D1
Current, Current Density,
Resistance and Ohm’s Law,
Magnetic Field, Magnetic Force
Today’s Reading Assignment: Current, Current
Density, and Resistance and Ohm’s Law,
Magnetic Fields and Forces Course Notes:
Sections 6.1-6.5, 8.1-8.3
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Announcements
Week 6 Problem Solving and Math Review Tuesday
from 9-11 pm in 26-152
PS 5 due Week 6 Tuesday at 9 pm in boxes outside
32-082 or 26-152
W06D2 Reading Assignment Course Notes: Magnetic
Forces, Currents & Dipoles; Sections 8.3, 9.1-9.2
Exam 2 Thursday March 21 7:30 - 9:30 pm
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Outline
Current and Current Density
Resistance and Ohm’s Law
Magnetic Field
Magnetic Forces
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Current: Flow Of Charge
Average current Iav: Charge DQ
flowing across area A in time Dt
DQ
I av =
Dt
Instantaneous current:
differential limit of Iav
dQ
I
dt
Units of Current: Coulomb/second = Ampere
4
How Big is an Ampere?
•
•
•
•
Household Electronics ~1 A
Battery Powered
~100 mA (1-10 A-Hr)
Household Service
100 A
Lightning Bolt
10 to 100 kA
• To hurt you
• To throw you
• To kill you
40 (5) mA DC(AC)
60 (15) mA DC(AC)
0.5 (0.1) A DC(AC)
• Fuse/Circuit Breaker
15-30 A
5
Direction of the Current
Direction of current is direction of flow of pos. charge
or, opposite direction of flow of negative charge
6
Why Does A Skydiver Fall At Constant
Speed (eventually)?
7
Why Does A Skydiver Fall At Constant
Speed (eventually)?
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Why Does A Skydiver Fall At Constant
Speed (eventually)?
When you first jump you are accelerating downward at g
As you build up speed, you encounter aerodynamic drag
(proportional to speed squared) which decreases your
acceleration
Eventually your downward speed is large enough that
the aero drag exactly balances downward force of
gravity, and you no longer accelerate
Then you move at constant speed, “terminal velocity”
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Why Does Current Flow?
If an electric field is set up in a conductor, charge
will move (making a current in direction of E)
Note that when current is flowing, the conductor is
not an equipotential surface (and Einside ≠ 0)!
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Microscopic Picture
Drift velocity is the average velocity forced by
applied electric field in the presence of collisions.
Magnitude is typically 4x10-5 m/sec, or 0.04
mm/second!
To go one meter at this speed takes about 10 hours!
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Summary
Current:
I
Charge
DQ = q(nADx)
Displacement
Drift speed
Dx = vd Dt
I avg
DQ
=
= nqvd A
Dt
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Current Density J
Let n = number of charged objects per unit volume
q = charge of object
v q = drift velocity of object
The current density is
current per unit area
J º nqv q Þ å ni qi v q
i
i
Generalization for many charged moving objects
J º å ni qi v q
i
i
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Current and Current Density J
J º nqv q Þ å ni qi v q
i
i
Current is the flow (flux) of
current density through an
open surface
I = ò J × n̂ dA = ò J × d A
S
Special case:
S
J
uniform and perpendicular to surface
I = JA
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Concept Question: Current Density
A current I = 200 mA flows in the wire below. What is
the magnitude of the current density J?
5 cm
10 cm
1.
2.
3.
4.
5.
6.
J = 40 mA/cm
J = 20 mA/cm
J = 10 mA/cm
J = 1 mA/cm2
J = 2 mA/cm2
J = 4 mA/cm2
20 cm
P18- 15
Concept Q. Answer: Current Density
Answer: 6. J = 4 mA/cm2
5
cm
20 cm
10
cm
The area that matters is the cross-sectional area
that the current is punching through – the 50 cm2
area shaded grey.
So:
J = I/A = 200 mA/50 cm2 = 4 mA/cm2
P18- 16
Conductivity and Resistivity
Ability of current to
flow depends on
density of charges &
rate of scattering
Two quantities summarize this:
σc: conductivity
 r: resistivity
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Microscopic Ohm’s Law
E = rr J
rr º
;
J = s cE
1
sc
rr and s c depend only on the microscopic
properties of the material, not on its shape
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Demonstrations:
Temperature Effects on Resistance F4
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%204&show=0
Conducting Glass F1
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%201&show=0
Conductivity of Ionizing Water F5
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%205&show=0
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Why Does Current Flow?
Instead of thinking of Electric Field, think of potential
difference across the conductor
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Ohm’s Law
What is relationship between electric potential
difference and current?
b
DV = Vb - Va = - ò E × d s = E
a
J=
E
r
I
J=
A
=
DV /
r
ü
ïï
ær ö
ý Þ DV = I ç ÷ º IR
è Aø
ï
ïþ
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Ohm’s Law
DV = IR
R=
r
A
R has units of Ohms (W) = Volts/Amp
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How Big is an Ohm?
• Short Copper Wire
milliohms (m Ω )
• Notebook paper (thru) ~1 G Ω
• Typical resistors
Ω to 100 M
Ω
• You (when dry)
100 k Ω
• You (when wet)
1kΩ
• Internally (hand to foot) 500 Ω
Stick your wet fingers in an electrical socket:
You’re dead!
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Concept Question: Resistance
When a current flows in a wire of length L and cross
sectional area A, the resistance of the wire is
1.
2.
3.
Proportional to A; inversely proportional to L.
Proportional to both A and L.
Proportional to L; inversely proportional to A.
4. Inversely proportional to both L and A
P18- 24
Concept Question Answer: Resistance
3. Proportional to L; inversely proportional to A.
The longer the wire the higher the resistance.
The bigger the cross-sectional area of the wire,
the more ways that current can flow through it,
so the lower the resistance.
So, if resistivity is rr , then
R=
rr L
A
P18- 25
Group Problem: Calculating Resistance
Consider a hollow cylinder of length L and inner
radius a and outer radius b. The material has resistivity
rr
.
Suppose a potential difference is
applied between the ends of the
cylinder and produces a current
flowing parallel to the axis. What
is the resistance measured?
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Magnetic Fields
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Magnetic Field of the Earth
North magnetic pole
located in southern hemisphere
http://www.youtube.com/watch?v=AtDAOxaJ4Ms
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Demonstrations:
Magnetic Field Lines of a Bar Magnet G2
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%202&show=0
Magnetic Field Lines of a Single Wire G12
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%2012&show=0
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How Big is a Tesla?
• Earth’s Field
5 x 10-5 T = 0.5 Gauss
• Brain (at scalp)
~1 fT
• Refrigerator Magnet
1 mT
• Inside MRI
3T
• Good NMR Magnet
18 T
• Biggest in Lab
150 T (pulsed)
• Biggest in Pulsars
108 T
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Moving Charges Feel Magnetic Force
FB = q v ´ B
Magnetic force perpendicular both to:
Velocity v of charge and magnetic field B
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Concept Question: Units Magnetic Field
What are the correct SI units for the magnetic
field?
1.
2.
3.
4.
5.
C/N-m-s
N-m-s/C
N/C
N-s/C-m
C-m/N-s
P18- 32
Concept Q. Answer: B Field Units
Fmag = q v ´ B
Since
newton
N
N
B Units =
=1
=1
C ×m s
A ×m
coulomb meter / second
(
)(
)
This is called 1 Tesla (T)
1 T = 104 Gauss (G)
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Cross Product: Magnitude
Computing magnitude of cross product A x B:
C= A´B
C=
A B sin q
| C |: area of parallelogram
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Cross Product: Direction
Right Hand Rule #1:
C= A´B
1) Curl fingers of right hand so that
you are moving A into B through the
smallest angle
2) Thumb will point in direction of C
35
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Cross Product: Direction
Right Hand Rule #1:
C= A´B
1) Curl fingers of right hand so that
you are moving A into B through the
smallest angle
2) Thumb will point in direction of C
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Cross Product: Signs
ˆi  ˆj  kˆ
ˆj  kˆ  ˆi
ˆj  ˆi  kˆ
kˆ  ˆj  ˆi
kˆ  ˆi  ˆj
ˆi  kˆ  ˆj
Cross Product is Cyclic (left column)
Anti-commutative:
Switching vectors changes sign (right column)
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Concept Question: Cross Product
An electron is traveling to the right with speed v in a
magnetic field that points up. What is the direction of
the force on the electron
v
1.
2.
3.
4.
5.
6.
up
down
left
right
into page
out of page
B
P18- 39
Concept Question Answer: Cross
Product and Magnetic Force
Answer: 5. v ´ B points out of the page but
the charge of the electron is negative
so FB = q v ´ B points into the page.
P18- 40
Demonstration:
Magnetic Deflection of TV Image
G6
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%206&show=0
41
Concept Question: Force Direction
Is this picture (deflection direction) correct?
1. Yes
2. No
3. I don’t know
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Concept Q. Answer: Force Direction
Answer: 1. Yes
Field from N to S, beam velocity right to left, cross
product is up. But charges are negative so force is
down, as pictured.
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Putting it Together: Lorentz Force
Force on charged particles in electric and
magnetic fields
Felec = qE
Fmag = q v ´ B
Electric Force
Magnetic Force
(
F= q E+v´B
)
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Velocity Selector
Particle moves in a straight line when
Fnet
E
= q(E + v ´ B) = 0 Þ v =
B
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What Kind of Motion in Uniform
B Field?
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Group Problem: Cyclotron Motion
A charged particle with charge
q is moving with speed v in
a uniform magnetic field B
as shown in the figure.
(1) Draw a sketch of the orbit
Find
(2) R : radius of the circle
(3) T : period of the motion
(4) w : cyclotron angular
frequency
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