W06D1 Current, Current Density, Resistance and Ohm s Law

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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

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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

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

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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

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Direction of the Current Direction of current is direction of flow of pos. charge

or, opposite direction of flow of negative charge

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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:

Charge

Displacement

Drift speed

I

ΔQ = q(nAΔx)

Δx = vdΔt

Iavg =

ΔQΔt

= nqvd A

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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∑

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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

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Conductivity and Resistivity

σc: conductivity ρr: resistivity

Ability of current to flow depends on density of charges & rate of scattering

Two quantities summarize this:

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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

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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



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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?

ΔV =Vb −Va = −

E ⋅ d s

a

b

∫ = E

J =Eρ=ΔV /

ρ

J =IA

⎫

⎬⎪⎪

⎭⎪⎪

⇒ ΔV = I ρA

⎛⎝⎜

⎞⎠⎟≡ IR

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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

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Drude Model: Conductivity

J = −nevdrift = ( ne2τ

me

)E

=σ c

E⇒

σ c =ne2τme

http://www.youtube.com/watch?v=dyX5I_io7bg

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Ohm’s Law

R =

ρA ΔV = IR

R has units of Ohms (W) = Volts/Amp

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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

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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

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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

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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 )

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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

Magnetic Field Lines of a Single Wire G12

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%202&show=0


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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

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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

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

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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 )

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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

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Demonstration: Magnetic Deflection of TV Image

G6


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Concept Question: Force Direction Is this picture (deflection direction) correct?

1.  Yes 2.  No 3.  I don’t know

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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

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Velocity Selector

Particle moves in a straight line when

Fnet = q(

E + v ×

B) = 0

⇒ v = E

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) : cyclotron angular frequency

ω

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W06D1 Current, Current Density, Resistance and Ohm s Law