View
11
Download
0
Category
Preview:
Citation preview
1
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, 8.5
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 20 7:30 - 9:30 pm
2
3
Outline
Current and Current Density Resistance and Ohm’s Law Magnetic Field Magnetic Forces
4
Current: Flow Of Charge
Units of Current: Coulomb/second = Ampere
Average current Iav: Charge flowing across area A in time
Iav =ΔQΔt
Instantaneous current: differential limit of Iav
dQIdt
=
ΔQ Δt
5
How Big is an Ampere? • Household Electronics • Battery Powered • Household Service • Lightning Bolt
• To hurt you • To throw you • To kill you
• Fuse/Circuit Breaker
~1 A ~100 mA (1-10 A-Hr) 100 A 10 to 100 kA
40 (5) mA DC(AC) 60 (15) mA DC(AC) 0.5 (0.1) A DC(AC)
15-30 A
6
Direction of the Current Direction of current is direction of flow of pos. charge
or, opposite direction of flow of negative charge
7
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)!
8
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!
9
Summary
Current:
Charge
Displacement
Drift speed
I
ΔQ = q(nAΔx)
Δx = vdΔt
Iavg =
ΔQΔt
= nqvd A
10
Current Density J Let n = number of charged objects per unit volume q = charge of object = drift velocity of object The current density is current per unit area Generalization for many charged moving objects
J ≡ nqvq ⇒ niqi
vqii∑
vq
J ≡ niqi
vqii∑
11
Current and Current Density J
J ≡ nqvq ⇒ niqi
vqii∑
Current is the flow (flux) of current density through an open surface Special case: uniform and perpendicular to surface
J
I =
J ⋅ ndA =
S∫
J ⋅dA
S∫
I = JA
P18- 12
Concept Question: Current Density A current I = 200 mA flows in the wire below. What is the magnitude of the current density J?
20 cm
10 cm 5 cm
1. J = 40 mA/cm 2. J = 20 mA/cm 3. J = 10 mA/cm 4. J = 1 mA/cm2
5. J = 2 mA/cm2
6. J = 4 mA/cm2
13
Conductivity and Resistivity
σc: conductivity ρr: resistivity
Ability of current to flow depends on density of charges & rate of scattering
Two quantities summarize this:
14
Microscopic Ohm’s Law
E = ρr
J
J =σ c
E
and depend only on the microscopic properties of the material, not on its shape
ρr ≡1σ c
ρr σ c
15
Demonstrations:
Temperature Effects on Resistance F4
Conducting Glass F1
Conductivity of Ionizing Water F5
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%204&show=0
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%205&show=0
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%201&show=0
16
Why Does Current Flow? Instead of thinking of Electric Field, think of potential difference across the conductor
17
Ohm’s Law What is relationship between electric potential difference and current?
ΔV =Vb −Va = −
E ⋅ d s
a
b
∫ = E
J =Eρ=ΔV /
ρ
J =IA
⎫
⎬⎪⎪
⎭⎪⎪
⇒ ΔV = I ρA
⎛⎝⎜
⎞⎠⎟≡ IR
18
Drude Model
Electrons scatter on average once every seconds. After every collision, direction of electron is random (hard sphere model)
Between collisions, electric field E gives each electron a drift momentum
If we average over all the electrons, then
the initial velocities before the collision are random and add to zero so the average velocity after the collision is
τ
me
vafter = me
vbefore + (−e)Eτ
(vbefore )ave =
0
vdrift ≡ (vafter )ave = (− eτme
)E
http://www.youtube.com/watch?v=dyX5I_io7bg
19
Drude Model: Conductivity
J = −nevdrift = ( ne2τ
me
)E
=σ c
E⇒
σ c =ne2τme
http://www.youtube.com/watch?v=dyX5I_io7bg
20
Ohm’s Law
R =
ρA ΔV = IR
R has units of Ohms (W) = Volts/Amp
21
How Big is an Ohm? • Short Copper Wire • Notebook paper (thru) • Typical resistors • You (when dry) • You (when wet) • Internally (hand to foot)
milliohms (m ) ~1 G to 100 M 100 k 1 k 500
Stick your wet fingers in an electrical socket:
I =V / R 120V / 1kΩ 0.1A You’re dead!
Ω
Ω Ω
Ω
Ω Ω
Ω
P18- 22
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. Proportional to A; inversely proportional to L. 2. Proportional to both A and L. 3. Proportional to L; inversely proportional to A. 4. Inversely proportional to both L and A
Worked Ex.: Calculating Resistance
23
Consider a hollow cylinder of length L and inner radius a and outer radius b. The material has resistivity .
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?
ρr
Worked Ex.: Calculating Resistance
24
Consider a hollow cylinder of length L and inner radius a and outer radius b. The material has resistivity .
When a potential difference is applied between the ends of the cylinder, current flows parallel to the axis. In this case, the cross-sectional area is
and the resistance is given by
ρr
A = π (b2 − a2 )
R =
ρr LA
=ρr L
π (b2 − a2 )
Group Problem: Calculating Resistance
25
Consider a material of resistivity in a shape of a truncated cone of altitude h, and radii a and b, for the right and the left ends, respectively, as shown in the figure. Assuming that the current is distributed uniformly throughout the cross-section of the cone, what is the resistance between the two ends? You may find the following integral useful (where and are constants).
ρr
!
du(αu + β )2∫ = − 1
α (αu + β )
26
Magnetic Fields
27
Magnetic Field of the Earth
North magnetic pole located in southern hemisphere
http://www.youtube.com/watch?v=AtDAOxaJ4Ms
28
Demonstrations:
Magnetic Field Lines of a Bar Magnet G2
Magnetic Field Lines of a Single Wire G12
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%202&show=0
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%2012&show=0
29
How Big is a Tesla? • Earth’s Field • Brain (at scalp) • Refrigerator Magnet • Inside MRI • Good NMR Magnet • Biggest in Lab • Biggest in Pulsars • LHC magnets 8.4 T • (27 km long)
5 x 10-5 T = 0.5 Gauss ~1 fT 1 mT 3 T 18 T 150 T (pulsed) 108 T
30
Moving Charges Feel Magnetic Force
FB = q v ×
B
Magnetic force perpendicular both to: Velocity v of charge and magnetic field B
P18- 31
Concept Question: Units Magnetic Field What are the correct SI units for the magnetic field?
1. C/N-m-s 2. N-m-s/C 3. N/C 4. N-s/C-m 5. C-m/N-s
32
Cross Product: Magnitude
Computing magnitude of cross product A x B:
C =A ×B
C =
AB sinθ
area of parallelogram |C |:
Vector Cross Product Magnitude: equal to the area of the parallelogram defined by the two vectors
Direction: determined by the Right-Hand-Rule
A ×B =
AB sinθ =
AB sinθ( ) =
A sinθ( ) B (0 ≤θ ≤ π )
34
Cross Product: Signs
jkijik
ijkikj
kijkji
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
−=×=×
−=×=×
−=×=×
Cross Product is Cyclic (left column) Anti-commutative: Switching vectors changes sign (right column)
P18- 35
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
1. up 2. down 3. left 4. right 5. into page 6. out of page
v
B
36
Demonstration: Magnetic Deflection of TV Image
G6
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%206&show=0
37
Concept Question: Force Direction Is this picture (deflection direction) correct?
1. Yes 2. No 3. I don’t know
38
Putting it Together: Lorentz Force
F = q
E + v ×
B( )
Fmag = q v ×
B
Felec = q
E
Force on charged particles in electric and magnetic fields
Electric Force Magnetic Force
39
Velocity Selector
Particle moves in a straight line when
Fnet = q(
E + v ×
B) = 0
⇒ v = E
B
40
What Kind of Motion in Uniform B Field?
41
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) : cyclotron angular frequency
ω
Recommended