Physics 202, Lecture 24Exam 3 Review

Logistics (November 30, 5:30-7PM; details see email) Topics: Inductance, AC Circuits, Waves, EM waves

(Chapters 28.6-28.9, 29, 30, and materials on waves)

Inductance:self inductance, mutual inductance, magnetic energy, RL circuit, …

AC Circuits:RC circuits,LC circuits, RLC circuits, impedance, phasers, …

Waves:transverse/longitudinal waves, wave function, sinusoidal waves, …

EM Waves: Maxwell’s equations,energy carried by EM waves, Poynting vector,

EM wave spectrum, antennas, …

The magnetic flux due to selfinductance is proportional to I:

The induced emf is proportionalto dI/dt:

When the current in a conducting device changes,an induced emf is produced in the opposite directionof the source current self inductance

Self Inductance

L: Inductance, unit: Henry (H)

Exercise: Calculate Inductance of a Solenoid

show that for an ideal solenoid:

(see board)

Reminder: magnetic field inside the solenoid�

L = µ0N2Al

Area: A

# of turns: N

Mutual Inductance For coupled coils:

ε2 = - M12 dI1/dt

ε1 = - M21 dI2/dt

Can prove (not here):M12=M21=M

M: mutual inductance(unit: also Henry)

ε2 = - M dI1/dt

ε1 = - M dI2/dt

Energy in an Inductor Energy stored in an inductor is U= 1/2 LI2

This energy is stored in the form of magnetic field: energy density: uB = 1/2B2/µ0 (recall: uE= 1/2 ε0E2)

Compare: Inductor: energy stored U= 1/2 LI2 Capacitor: energy stored U= 1/2 C(ΔV)2

Resistor: no energy stored, (all energy converted to heat)

RL Circuit

The time constant is τ=L/R

B

E

C�

L = µ0N2Al

U= 1/2 LI2

B

C

A

(4 Pi 10^{-7} * 200/0.5) *(Pi * (0.05)^2)*200

LC Circuit and Oscillation Exercise: Find the oscillation frequency of a LC circuit

0)(/)( =−−dttdILctq

)sin()cos(

1

0)(1)(

max

max

2

2

2

φωωφω

ω

ω

+−=+=

=

=+

tQItQq

LC

dttqdtq

0)(/)( 2

2

=+dttqdLCtq

eq. of Harmonic Oscillation

Total Energy is conserved

I

Current And Voltages in a Series RLC Circuit

ΔVmax Sin(ωt +φ)

i= Imax Sin(ωt)

ΔvR=(ΔVR)max Sin(ωt)ΔvL=(ΔVL)max Sin(ωt + π/2)ΔvC=(ΔVC)max Sin(ωt - π/2)

22maxmax )( CL XXRIV −+=Δ

)(tan 1

RXX CL −= −φ

Quiz: Can the voltage amplitudes across each components , (ΔVR)max , (ΔVL)max , (ΔVC)max larger thanthe overall voltage amplitude ΔVmax ?

�

ΔV = ΔVmax sin(ωt + φ)

Impedance For general circuit configuration: ΔV=ΔVmaxsin(wt+φ) , ΔVmax=ImaxZ Z: is called Impedance.

e.g. RLC circuit :

In general impedance is a complex number. Z=Zeiφ. It can be shown that impedance in series and parallel

circuits follows the same rule as resistors. Z=Z1+Z2+Z3+… (in series) 1/Z = 1/Z1+ 1/Z2+1/Z3+… (in parallel)(All impedances here are complex numbers)

ΔV

i

22 )( CL XXRZ −+=

ZC

)sin()cos( tNABdt

ABdNdtdN ωωθε =−=Φ−= B

P1/V+P2/V=20A B

f=1/T E

Ζ=1/(ωC) C

A

ω=ω0 = LC1

A

22 )( CL XXRZ −+=

B

Transverse and Longitudinal Waves If the direction of mechanic disturbance

(displacement) is perpendicular to the direction ofwave motion, the wave is called transverse wave.

If the direction of mechanic disturbance (displacement)is parallel to the direction of wave motion, the wave iscalled longitudinal wave.

EM waves are always transverse.

Linear Wave Equation Linear wave equation

2

2

22

2 1ty

vxy

∂∂=

∂∂

certain physical quantity

Wave speed

Sinusoidal wave

)22sin( φπλπ +−= ftxAy

A:Amplitude

f: frequency φ:Phase

General wave: superposition of sinusoidal waves

v=λfk=2π/λω=2πf

λ:wavelength

Maxwell Equations

�

E • dA = qε0

∫

�

E • d = − dΦB

dt∫

�

F = qE + qv ×B

�

B• d = µ0I + ε0µ0dΦE

dt∫�

B• dA = 0∫

Gauss’s Law/ Coulomb’s Law

Faraday’s Law

Gauss’s Law of Magnetism, no magnetic charge

Ampere Maxwell Law

Also, Lorentz force Law

These are the foundations of the electromagnetism

Properties of EM Waves EM waves are transverse waves traveling at speed c:

Recall: energy densities uE= 1/2 ε0E2, uB= 1/2 B2/µ0 For a EM wave, at any time/location, uE = 1/2 ε0E2 = 1/2 B2/µ0 =uB (using E/B=c)

In an EM wave, the energies carried by electricfield and magnetic field are always the same.

Total energy stored (per unit of volume): u=uE+uB = ε0E2 = B2/µ0 Power transmitted per unit of area is equal to uc in the direction of wave Averaging over time: uav= 1/2 ε0Emax

2 = 1/2 Bmax2/µ0 , uavc = I (intensity)

x

y

z

EB

2

2

002

2

tE

xE yy

∂∂

=∂∂

εµ2

2

002

2

tB

xB zz

∂∂=

∂∂ εµ

The Poynting Vector The rate of flow of energy in an em

wave is described by a vector, S ,called the Poynting vector

The Poynting vector is defined as

Its direction is the direction ofpropagation

This is time dependent Its magnitude varies in time Its magnitude reaches a

maximum at the same instant asE and B

�

S ≡ 1

µ0

E × B

Power per unit of area�

I = Sav

Antennas Antennas are essentially arrangement of conductors for

transmitting and receiving radio waves. Parameters: gain, impedance, frequency, orientation,

polarization, etc.

half-wave antenna

loop

λ/4

λ/4

maximumstrength

mocristrip

E

C

uav= 1/2 ε0Emax2 = 1/2 Bmax

2/µ0 , uavc = I

Erms2 =1/2 Emax

2 D

D

)sin()cos( tNABdt

ABdNdtdN ωωθε =−=Φ−= B

E=cB

I=Power/area A

A

None of the above: 12 mT