Tue. Nov. 9, 2009Physics 208, Lecture 201 From last time… Inductors in circuits Inductors Flux = (Inductance) X (Current)

Tue. Nov. 9, 2009 Physics 208, Lecture 20 1

From last time…

Inductors in circuits

InductorsFlux = (Inductance) X (Current)

€

Φ=LI

Vbatt

R

L

IVb

Va

Voltage drop across inductor Constant current

No voltage difference Current changing in time

Voltage difference across inductor


€

ΔVL = Vb −Va = −LdI

dt

RL Circuit

What is voltage across L just after switch closed?


Before switch closed, IL = 0 Current through inductor cannot ‘jump’ Just after switch closed, IL= 0.

€

VL = −LdI

dt

A. VL = 0

B. VL= Vbattery

C. VL= Vbattery / R

D. VL= Vbattery / L

Kirchoff’s loop law:

VR + VL = Vbattery

R and L in series, IL=0 IR=0, VR=0


€

VL t = 0( ) = −Vbattery = −LdIL

dt

⇒dIL

dt=

Vbattery

L

IL

IL(t)

Time ( t )

00

Slope dI / dt = Vbattery / L

€

IL t = 0( ) = 0

IL instantaneously zero, but increasing in time

Just a little later…

A short time later ( t=0+Δt ), the current is increasing …


IL(t)

Time ( t )0

0

Slope dI / dt = Vbattery / L

A. More slowly

B. More quickly

C. At the same rate

IL>0, and IR=IL

VR≠0, so VL smaller

VL= -LdI/dt, so dI/dt smaller

Switch closed at t=0


IL

IL(t)

Time ( t )0

0

Initial slope

What is current through inductor in equilibrium, a long time after switch is closed?

Later slope

€

dI

dt=

Vbattery

L€

dI

dt=

Vbattery

L− IL t = Δt( )R

A. Zero

B. Vbattery / L

C. Vbattery / R

Equilibrium: currents not changing

dIL / dt =0, so VL=0

VR=Vbattery

IL = IR =Vbattery / R

€

Vbattery

R


RL summary

I(t)

€

I t( ) = I∞ 1− e−t /(L / R )( ) = I∞ 1− e−t /τ

( )

€

I∞ = Vbattery /R

τ = L /R = time constant

I(t)

Switch closed at t=0

€

I∞ = Vbattery /R

Question

What is the current through R1 immediately after the switch is closed?


L

R1

R2

A. Vbattery / L

B. Vbattery / R1

C. Vbattery / R2

D. Vbattery / (R1+R2)

E. 0

IL cannot ‘jump’. IL=0 just after closing switch.

All current flows through resistors.

Resistor current can jump.

Thinking about electromagnetism

Many similarities between electricity, magnetism Some symmetries, particularly in time-dependence


Electric Fields

Arise from charges Capacitor, Q=CV

Arise from time-varying B-field Inductor, Faraday effect

Magnetic Fields

Arise from currents Inductor, Φ=LI

Arise from time-varying E-field


Maxwell’s unification Intimate connection

between electricity and magnetism

Time-varying magnetic field induces an electric field (Faraday’s Law)

Time-varying electric field generates a magnetic field

This is the basis of Maxwell’s unification of electricity and magnetism into Electromagnetism

€

r∇ ×

rE = −

1

c

∂r B

∂tr

∇ ×r B =

1

c

∂r E

∂t

In vacuum:


•A Transverse wave.

•Electric/magnetic fields perpendicular to propagation direction

•Can travel in empty space

f = v/, v = c = 3 x 108 m/s (186,000 miles/second)


The EMSpectrum

Types are distinguished by frequency or wavelength

Visible light is a small portion of the spectrum


Sizes of EM waves Visible light

typical wavelength of 500 nm = = 0.5 x 10-6 m = 0.5 microns (µm)

AM 1310, your badger radio network, has a vibration frequency of 1310 KHz = 1.31x106 Hz

What is its wavelength?A. 230 m

B. 0.044 m

C. 2.3 m

D. 44m


A microwave oven irradiates food with electromagnetic radiation that has a frequency of about 1010 Hz. The wavelengths of these microwaves are on the order of

A. kilometers

B. meters

C. centimeters

D. micrometers

Quick Quiz


Mathematical description

€

Bo = Eo /c

€

rE =

r E o cos kz −ωt( )

r B =

r B o cos kz −ωt( )

€

rE ⊥

r B

€

k =2π

λ, ω = 2πf

z

x

y

Propagation direction =

€

rE ×

r B


EM Waves from an Antenna

Two rods are connected to an ac source, charges oscillate between the rods (a)

As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b)

The charges and field reverse (c) The oscillations continue (d)


Detecting EM waves

FM antenna AM antenna

Oriented vertically for radio waves


Transatlantic signals

Gulgielmo Marconi’s transatlantic transmitter

Capacitor banksInduction

coilsSpark gap


Transatlantic receiver

Left to right: Kemp, Marconi, and Paget pose in front of a kite that was used to keep aloft the receiving aerial wire used in the transatlantic radio experiment.


Energy and EM Waves

Energy density in E-field

Energy density in B-field

€

uE = εoE 2 r, t( ) /2

€

uB = B2 r, t( ) /2μo

Total

€

uTot = εoE 2 /2 + B2 /2μo

= εoE 2 /2 + E 2 /2c 2μo = εoE 2 r, t( ) = B2 r, t( ) /μo

€

uTot = εoE 2 = εoEo2 cos2 kz −ωt( ) moves w/ EM wave

at speed c


Power and intensity in EM waves Energy density uE moves at c

Instantaneous energy flow = energy per second passing plane = This is power density W/m2

Oscillates in time Time average of this is Intensity =

€

cεoEmax2 /2 = cBmax

2 /2μo€

cuTot = cεoE 2 = cεoEo2 cos2 ωt( )


Example: E-field in laser pointer

3 mW laser pointer.

Beam diameter at board ~ 2mm

Intensity =

€

10−3W

π 0.001m( )2 = 318W /m2

How big is max E-field?

€

cεoEmax2 /2 = 318W /m2

Emax =2 318W /m2

( )

3×108 m /s( ) 8.85 ×10−12C2 /N ⋅m2( )

= 489N /C = 489V /m


Spherical waves Sources often radiate EM wave in all directions

Light bulb The sun Radio/tv transmission tower

Spherical wave, looks like plane wave far away Intensity decreases with distance

Power spread over larger area

€

I =Psource

4π r2

Source power

Spread over thissurface area


QuestionA radio station transmits 50kW of power from its

antanna. What is the amplitude of the electric field at your radio, 1km away.

€

I =50,000W

4π 1000m( )2 = 4 ×10−3W / m2

€

cεoEmax2 /2 = 4 ×10−3W /m2

Emax =2 4 ×10−3W /m2( )

3 ×108 m /s( ) 8.85 ×10−12C2 /N ⋅m2( )

=1.73N /C =1.73V /m

A. 0.1 V/m

B. 0.5 V/m

C. 1 V/m

D. 1.7 V/m

E. 15 V/m


The Poynting Vector Rate at which energy flows through a unit area perpendicular

to direction of wave propagation

Instantaneous power per unit area (J/s.m2 = W/m2) is also

Its direction is the direction of propagation of the EM wave

This is time dependent Its magnitude varies in time Its magnitude reaches a maximum at the

same instant as E and B

€

rS =

1

μo

r E ×

r B ≡ Poynting Vector


Radiation Pressure Saw EM waves carry energy They also have momentum When object absorbs energy U from EM wave:

Momentum Δp is transferred

Result is a force

Pressure = Force/Area = €

Δp = U /c ( Will see this later in QM )

€

F = Δp /Δt =U /Δt

c= P /c

€

prad =P / A

c= I /c

Radiation pressure on perfectly absorbing object

Power

Intensity


Radiation pressure & force

EM wave incident on surface exerts a radiation pressure prad (force/area) proportional to intensity I.

Perfectly absorbing (black) surface:

Perfectly reflecting (mirror) surface:

Resulting force = (radiation pressure) x (area) €

prad = I /c

€

prad = 2I /c


QuestionA perfectly reflecting square solar sail is 107m X 107m. It has

a mass of 100kg. It starts from rest near the Earth’s orbit, where the sun’s EM radiation has an intensity of 1300 W/m2.

How fast is it moving after 1 hour?

€

prad = 2I /c

Frad = prad A = 2IA /c =2 1300W /m2( ) 1.145 ×104 m2

( )

3 ×108 m /s= 0.1N

a = Frad /m =10−3 m /s2

v = at = 10−3 m /s2( ) 3600s( ) = 3.6m /s

A. 100 m/s

B. 56 m/s

C. 17 m/s

D. 3.6 m/s

E. 0.7 m/s

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Tue. Nov. 9, 2009Physics 208, Lecture 201 From last time… Inductors in circuits Inductors Flux = (Inductance) X (Current)