Electricity and Magnetism • Review for Quiz #3 –Exp EB – in B-Field • on free charge • on current – Sources of B-Field •Biot-Savart • ’s Law –Exp MF – Magnetic Induction • Faradays Law •Lenz’Rule

Apr 5 2002 Electrical Breakdown

• Need lot’s of free charges • But stuck in potential well of nucleus U • Need energy ∆U to jump out of well • How to provide this energy?

e- ∆U

Apr 5 2002 Impact Ionization

e- Ukin > ∆U

E

• Define Vion = ∆U/q Ionization potential

•One e- in, two e- out E • Avalanche?

Apr 5 2002 Impact Ionization (2) (1) - λmfp e λmfp : Mean Free Path

E • To get avalanche we need:

∆Ukin between collisions (1) and (2) > Vion * e • Acceleration in uniform Field

∆Ukin = V2 –V1 = e E d12 • Avalanche condition then

E > Vion /λmfp

Apr 5 2002 Impact Ionization

How big is Mean Free Path?

(i) If Density n is big -> λmfp small

(ii) If size σ of molecules is big -> λmfp small

λmfp = 1/(n σ )

Avalanche if E > Vion /λmfp = Vion n σ

Apr 5 2002 Magnetism

• Observed New Force between –two – and Iron – Magnet and wire carrying current – Wire carrying current and Magnet – Two wires carrying currents • Currents (moving charges) can be subject to and source of Magnetic Force

Apr 5 2002 Magnetic Force • Force between Magnets • Unlike Poles attract F F N S N S

• Like Poles repel F F N S SN

Apr 5 2002 Magnetic Force

• Magnets also attract non-magnets!

F Iron nail

Apr 5 2002 Current and Magnet

Wire, I = 0 Wire, I > 0 N N S X S

Apr 5 2002 Magnet and Current

F Wire I F

• Force on wire if I != 0 • Direction of Force depends on Sign of I • Force perpendicular to I

Apr 5 2002 Current and Current

I I I I

F F F F

Attraction Repulsion

Experiment MF

Apr 5 2002 Force on moving charge

-q FE v x B E FB R

FL = q E + q v x B R = m v/(q B) Lorentz-Force Cyclotron Radius

Apr 5 2002 Work done on moving charge

-q FE v x B dW = FB dL E FB = (q v x B) dL L = (q dL/dt x B) dL W = F L E = 0 = q E L does no Work!

Apr 5 2002 Force on Wire carrying current I

dF = dq v x B x B I B Wire = dq dL/dt x B dL = I dL x B

Apr 5 2002 Currents and B-Field

I • Current as Source of B • Magnetic Field lines are always closed – no Magnetic Charge (Monopole) • Right Hand Rule

Apr 5 2002 Currents and B-Field

I I

• Superposition Principle!

Apr 5 2002 Currents and B-Field

Ix x x x

• Solenoid: Large, uniform B inside • Superposition Principle!

Apr 5 2002 Magnetic Field vs Electric Field

B E r v +Q +Q

2 2 B = µ0/(4 π) Q/r v x r E = 1/(4 πε0) Q/r r

-7 -12 2 2 µ0 = 4 π 10 T m /A ε0 = 8.85 10 C /(Nm ) 8 2 2 1/(µ0 ε0) = (3 10 m/s) = c Speed of Light Deep connection between B and E Field

Apr 5 2002 Magnetic Field for Current I

2 dB = µ0/(4 π) dQ/r v x r for charge dQ I = dQ/dt -> dQ v = dQ dl/dt = I dl

2 Law of Biot-Savart dB = µ0/(4 π) I dl x r/r

Magnetic Field dB for current through segment dl

For total B-Field: Integrate over all segments dl

Apr 5 2002 Magnetic Field for Current I y

+L dl r dl φ x r I ν

-L z

Apr 5 2002 Magnetic Field for Current I

•For quiz: – No long calculations – But need to understand how to use Biot- Savart to find direction of B

Apr 5 2002 Remember: Gauss’ Law

Electric φE

Integral over any Electric Charge is the closed surface Source of Electric Field

Apr 5 2002 Gauss’ Law for Magnetic Fields

• Magnetic Flux through closed surface is 0 • This says: There are no magnetic monopoles • Important Law – one of Maxwell’s equations • Unfortunately of limited practical use

Apr 5 2002 Ampere’s Law I • Ampere’s idea: Relate Field B to its Source: I B Closed Line instead of closed surface!

Apr 5 2002 Ampere’s Law I Ampere’s Law helps because we can choose integration path!

B

Right-Hand rule for relating sign of dl and I

Apr 5 2002 Ampere’s Law I Ampere’s Law helps because we can choose integration path!

B

Apr 5 2002 Field of a Solenoid

x x x x x x x x • Current I • n turns per unit length a B b • (infinite length)

h B = µ0 I n

Loop C

c d L

Apr 5 2002 Coaxial Cable

Conductor I2

I1

Insulator

Outside field vanishes for I2 = I1

Apr 5 2002 Cylindrical Conductor

I • Uniform Current-Density J •Radius R r • J = I/(π R2) R

Apr 5 2002 Magnetic Induction

• Currents give rise to B-Field • Q: Can B-Field give rise to current? • A: Only if Magnetic Flux changes with time! • Took a very long time to realize...

Apr 5 2002 Faradays Law

Magnetic Flux (usually, A not closed surface)

Faradays Law

Apr 5 2002 Faradays Law

• ΦB can change because –|B| changes – Angle between B and A changes – |A| (size of circuit in B) changes

Apr 5 2002 Faradays Law

x B Moving circuit: Induced EMF is v consequence of force F on moving charges +q

Apr 5 2002 Lenz’ Rule

Lenz’ Rule:

Sign of Iind such that it opposes the flux change that generated it

Apr 5 2002 Use of Faradays Law

•To find Iind: x B

•Calculate ΦB • Find, what makes ΦB v change

• Find sign of Iind using Lenz’ rule

Apr 5 2002 Use of Faradays Law

x B • Sign of current: v Opposing change of ΦB -> Reducing B

Iind

Lenz’ Rule: Effect of Iind current opposing dΦB/dt is like ‘drag’ or ‘inertia’

Apr 5 2002 My favorite Demo

Falling Al disk B

• Falling Al ring is slowed down in B-Field • Induced Eddy-currents • Energy converted to heat Apr 5 2002 Faradays Law

x B Moving circuit: Induced EMF is v consequence of force on F moving charges +q

What about changing B?

Apr 5 2002