Current in an LC Circuit

LCtQQ cos0

dtdQI

LCt

LCQI sin0

Current in an LC circuit

Period: LCT 2

Frequency: f 1 / 2 LC

Current in an LC Circuit

Question (Chap. 23)

A. There will be no current in the circuit at any time because of the opposing emf in the inductor.

B. The current in the circuit will maximize at time t when the capacitor will have charge Q(t)=0.

C. The current in the circuit will maximize at time t when capacitor will have full charge Q(t)=Q0.

D. The current will decay exponentially.

Q0

A capacitor C was charged and contains charge +Q0 and –Q0 on each of its plates, respectively. It is then connected to an inductor (coil) L. Assuming the ideal case (wires have no resistance) which is true?

QuestionTwo metal rings lie side-by-side on a table. Current in the left ring runs clockwise and is increasing with time. This induces a current in the right ring. This current runs

A) ClockwiseB) Counterclockwise

when viewed from above

Single home current: 100 A serviceVwires=IRwires

Transformer: emfHV IHV = emfhomeIhome

Single home current in HV: <0.1 APower loss in wires ~ I2

Faraday’s Law: Applications

Faraday’s Law: Applications

Inductionmicrophone

Faraday’s Law: Applications

Chapter 24

Classical Theory of Electromagnetic Radiation

Maxwell’s Equations

0

ˆ

insideqdAnE

pathinsideIldB _0

Gauss’s law for electricity

Gauss’s law for magnetism

Complete Faraday’s law

Ampere’s law(Incomplete Ampere-Maxwell law)

0ˆ AnB

∮𝐸 ∙𝑑 𝑙=− 𝑑𝑑𝑡 [𝐵 ∙ ��𝑑 𝐴 ]

No current inside

0 ldB

Current pierces surface

IldB 0

rIB 2

40

IrrIldB 0

0 224

pathinsideIldB _0Ampere’s Law

Time varying magnetic field leads to curly electric field.

Time varying electric field leads to curly magnetic field?

dAnEelec ˆ

00

0cosQA

AQ

elec

dtdQ

dtd elec

0

1

I

0

1

I

dtdI elec

0 ‘equivalent’ current

pathinsideIldB _0 combine with current in Ampere’s law

Maxwell’s Approach

dtdIldB elec

pathinside 0_0

Works!

The Ampere-Maxwell Law

Four equations (integral form) :

Gauss’s law

Gauss’s law for magnetism

Faraday’s law

Ampere-Maxwell law

0

ˆ

insideqdAnE

dAnBdtdldE ˆ

dtdIldB elec

pathinside 0_0

+ Lorentz force BvqEqF

Maxwell’s Equations

0ˆ AnB

Time varying magnetic field makes electric field

Time varying electric field makes magnetic field

Do we need any charges around to sustain the fields?

Is it possible to create such a time varying field configuration which is consistent with Maxwell’s equation?

Solution plan: • Propose particular configuration• Check if it is consistent with Maxwell’s eqs• Show the way to produce such field• Identify the effects such field will have on matter• Analyze phenomena involving such fields

Fields Without Charges

Key idea: Fields travel in space at certain speedDisturbance moving in space – a wave?

1. Simplest case: a pulse (moving slab)

A Simple Configuration of Traveling Fields

0

ˆ

insideqdAnE

0ˆdAnE

Pulse is consistent with Gauss’s law

0ˆ AnB

Pulse is consistent with Gauss’s law for magnetism

A Pulse and Gauss’s Laws

dtd

emf mag

Since pulse is ‘moving’, B depends on time and thus causes E

Area doesnot move

tBhvmag

Bhvdt

dt

magmag

emf

EhldEemf

E=Bv

Is direction right?

A Pulse and Faraday’s Law

dtdIldB elec

pathinside 0_0

=0

tEhvelec

Ehvdt

dt

elecelec

BhldB

EvhBh 00

vEB 00

A Pulse and Ampere-Maxwell Law

vEB 00 E=Bv

vBvB 00

2001 v

m/s 8

00

1031

v

Based on Maxwell’s equations, pulse must propagate at speed of light

E=cB

A Pulse: Speed of Propagation

Question

At this instant, the magnetic flux mag through the entire rectangle is:

A) B; B) Bx; C) Bwh; D) Bxh; E) Bvh

Question

In a time t, what is mag?

A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h

Question

emf = mag/t = ?

A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h

Question

What is around the full rectangular path?

A) Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2Evt; E)2Evt

Question

emf dmag

dtBvh

rEgd

rl Eh—

What is E?

A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x

Exercise

If the magnetic field in a particular pulse has a magnitude of 1x10-5 tesla (comparable to the Earth’s magnetic field), what is the magnitude of the associated electric field?

E cB

Force on charge q moving with velocity v perpendicular to B:

E 3x108 m / s 1x10 5 T 3000V / m

𝐹𝑚𝑎𝑔

𝐹𝑒𝑙=𝑣𝐵

𝐸¿𝑣𝐵𝑐𝐵=

𝑣𝑐

Direction of speed is given by vector product BE

Direction of Propagation

Electromagnetic pulse can propagate in spaceHow can we initiate such a pulse?

Short pulse of transverseelectric field

Accelerated Charges

1. Transverse pulse propagates at speed of light

2. Since E(t) there must be B

3. Direction of v is given by: BE

E

Bv

Accelerated Charges

We can qualitatively predict the direction.What is the magnitude?

Magnitude can be derived from Gauss’s law

Field ~ -qa

rcaqEradiative 2

041

1. The direction of the field is opposite to qa

2. The electric field falls off at a rate 1/r

Magnitude of the Transverse Electric Field

An electron is briefly accelerated in the direction shown. Draw the electric and magnetic vectors of radiative field.

1. The direction of the field is opposite to qa

a

E

BE

2. The direction of propagation is given by

B

Exercise

Circular motion: Is there radiation emitted? va

Classical physics says “YES”Þ orbiting particle must lose energy!Þ speed decreasesÞ particle comes closer to center

Classical model of atom:Electrons should fall on nucleus!

To explain the facts - introduction ofquantum mechanics:Electrons can move around certain orbits only and emit E/M radiation only when jumping from one orbit to another

Stability of Atoms

fT

f

/12

Acceleration:

tydt

yda sin2max2

2

rcaqEradiative 2

041

jsin4

12

2max

0

trc

qyEradiative

Sinusoidal E/M field

Sinusoidal Electromagnetic Radiation

Sinusoidal E/M Radiation: Wavelength

fT

f

/12

Freeze picture in time:

Instead of period canuse wavelength:

cTfc

Example of sinusoidal E/M radiation:

atomsradio stationsE/M noise from AC wires

Electromagnetic Spectrum

Need to create oscillating motion of electrons

Radio frequencyLC circuit: can produce oscillating motion of chargesTo increase effect: connect to antenna

Visible lightHeat up atoms, atomic vibration can reach visible frequency rangeTransitions of electrons between different quantized levels

E/M Radiation TransmittersHow can we produce electromagnetic radiation of a desired frequency?

AC voltage(~300 MHz)

nolight

E/M radiation can be polarized along one axis…

…and it can be unpolarized:

Polarized E/M Radiation

Making polarized light Turning polarization

Polaroid sunglasses and camera filters:

reflected light is highly polarized: can block it

Considered: using polarized car lights and polarizers-windshields

Polarized Light