Physics 1202 Lecture 14 Todays Agenda Announcements Team

Physics 1202: Lecture 14 Today’s Agenda • Announcements: Team problems for. Thursday – Team 7: Cailin Catarina, Matthew Canapetti, Kervin Vincent – Team 8: Natalie Kasir, Adam Antunes, Quincy Alexander – Team 9: Garrett Schlegel, Joyce Nieh, Matthew Lombardo • Office hours: Monday 2: 30 -3: 30 Thursday 3: 00 -4: 00 • Homework #6: due this coming Friday • Midterm 1: – Solutions in class – Midterm sample + To-Know sheet on web already • Chapter 25: E&M waves – Production and properties

R C e ~ L

24 -5 RLC Circuits • Phasor diagram – useful to analyze an RLC circuit • Follow the loop • Total V V= VR + VL+ VC = Z Imax – VR = R Imax (in phase) – VL= XL Imax (leads by 90 o) – VC= XC Imax (lags by 90 o)

24 -5 RLC Circuits • The phase angle for an RLC circuit is: • If XL = XC, the phase angle is zero, and the voltage and current are in phase. • The power factor:

24 -6: Resonance - LC Circuits • Consider the LC and RC series circuits shown: C R C L • Suppose that the circuits are formed at t=0 with the capacitor C charged to a value Q. Claim is that there is a qualitative difference in the time development of the currents produced in these two cases. Why? ? • Consider from point of view of energy! • In the RC circuit, any current developed will cause energy to be dissipated in the resistor. • In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!

RC/LC Circuits i i Q +++ Q C +++ --- C R LC: current oscillates RC: current decays exponentially 0 i -i 0 0 1 t L t

The energy in LC circuit conserved ! When the capacitor is fully charged: When the current is at maximum (Io): The maximum energy stored in the capacitor and in the inductor are the same: At any time:

Resonance • For fixed R, C, L the current im will be a maximum at the resonant frequency 0 which makes the impedance Z purely resistive. ie: reaches a maximum when: XL=XC the frequency at which this condition is obtained is given from: Þ • Note that this resonant frequency is identical to the natural frequency of the LC circuit by itself! • At this frequency, the current and the driving voltage are in phase!

24 -6 Resonance in Electrical Circuits • In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency: XL=XC the frequency at which this condition is obtained is given from: Þ

Resonance The current in an LCR circuit depends on the values of the elements and on the driving frequency through the relation Z | XL-XC | | R “ Impedance Triangle” Suppose you plot the current versus , the source voltage frequency, you would get: m / R 0 R=Ro im 0 R=2 Ro 0 1 x 2 2 o

Application of Resonance o Tuning a radio o A varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received o Metal Detector o The portal is an inductor, and the resonant frequency is set to a condition with no metal present o When metal is present, it changes the effective inductance, which changes the current o The change in current is detected an alarm sounds

Fields from Circuits? • We have been focusing on what happens within the circuits we have been studying (eg currents, voltages, etc. ) • What’s happening outside the circuits? ? – We know that: » charges create electric fields and » moving charges (currents) create magnetic fields. – Can we detect these fields? – Demos: » We saw a bulb connected to a loop glow when the loop came near a solenoidal magnet. » Light spreads out and makes interference patterns. Do we understand this?

Chap. 25 f( x ) x x z y

Maxwell’s Predictions • Maxwell thought that electric and magnetic fields play symmetric roles in nature • Since a varying B induces an emf and hence E (Faraday’s Law), he hypothesized that a changing E would produce a B o field o • Maxwell calculated the speed of o light to be 3 x 108 m/s • He concluded that visible light and all other electromagnetic waves consist of fluctuating E and B fields, with each varying field inducing the other James Clerk Maxwell 1831 – 1879 Electricity and magnetism were originally thought to be unrelated In 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena Light: electromagnetic waves

Maxwell’s Equations • These equations describe all of Electricity and Magnetism. • They are consistent with modern ideas such as relativity. • They describe light ! (electromagnetic wave)

Hertz’s Confirmation of Maxwell’s Predictions o o Hertz measured the speed of the waves from the transmitter • He used the waves to form an interference pattern and calculated the wavelength • From v = f λ, v was found • v was very close to 8 • 3 x 10 m/s, the known speed of light This provided evidence in support of Maxwell’s theory • 1857 – 1894 • First to generate and detect electromagnetic waves in a laboratory setting • Showed radio waves could be reflected, refracted and diffracted • The unit Hz is named for him

25 -1: Generating E-M Waves • Static charges produce a constant Electric Field but no Magnetic Field. • Moving charges (currents) produce both a possibly changing electric field and a static magnetic field. • Accelerated charges produce EM radiation (oscillating electric and magnetic fields). • Antennas are often used to produce EM waves in a controlled manner.

• A Dipole Antenna V(t)=Vocos( t) + + - E E + + - • time t=0 x z y • time t= /2 • time t= / one half cycle later

EM Waves Produced by Antenna • Two rods are connected to an ac source, charges oscillate between o the rods (a) • As oscillations continue, the rods become less charged, the field near the charges decreases and the field o produced at t = 0 moves away from the rod (b) • The charges and field reverse (c) o • The oscillations continue (d) Because the oscillating charges in the rod produce a current, there is also a magnetic field generated As the current changes, the magnetic field spreads out from the antenna The magnetic field is perpendicular to the electric field

Electromagnetic Waves are Transverse Waves • The E and B fields are perpendicular to each other • Both fields are perpendicular to the direction of motion • Therefore, em waves are transverse waves o The ratio of the electric field to the magnetic field is equal to the speed of light

E & B in Electromagnetic Wave • Plane Harmonic Wave: where: y x z Note: the direction of propagation where is given by the cross product are the unit vectors in the (E, B) directions. Nothing special about (Ey, Bz); eg could have (Ey, -Bx) Note cyclical relation:

Lecture 14, ACT 1 • Suppose the electric field in an e-m wave is given by: – In what direction is this wave traveling ? (a) + z direction (c) +y direction (b) -z direction (d) -y direction

Lecture 14, ACT 2 • Suppose the electric field in an e-m wave is given by: • Which of the following expressions describes the magnetic field associated with this wave? (a) Bx = -(Eo/c)cos(kz + t) (b) Bx = +(Eo/c)cos(kz - t) (c) Bx = +(Eo/c)sin(kz - t)

Electromagnetic Waves produced by dipole • Any time an electric charge is accelerated, it will radiate: • Accelerated charges radiate electromagnetic waves.

Dipole radiation pattern proportional to sin( t) • oscillating electric dipole generates e-m radiation that is polarized in the direction of the dipole • radiation pattern is doughnut shaped & outward traveling – zero amplitude directly above and below dipole – maximum amplitude in-plane

E & B Fields in EM wave • How it looks like …

Receiving E-M Radiation receiving antenna y Speaker x z One way to receive an EM signal is to use the same sort of antenna. • Receiving antenna has charges which are accelerated by the E field of the EM wave. • The acceleration of charges is the same thing as an EMF. Thus a voltage signal is created.

Lecture 17, ACT 3 • Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground. y x z In which orientation should you turn your receiving dipole antenna in order to best receive this signal? a) Along S b) Along B C) Along E

Loop Antennas Magnetic Dipole Antennas • The electric dipole antenna makes use of the basic electric force on a charged particle • Note that you can calculate the related magnetic field using Ampere’s Law. • We can also make an antenna that produces magnetic fields that look like a magnetic dipole, i. e. a loop of wire. • This loop can receive signals by exploiting Faraday’s Law. For a changing B field through a fixed loop of area A: FB= A B

Lecture 14, ACT 4 • Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground. y x z In which orientation should you turn your receiving loop antenna in order to best receive this signal? a) â Along S b) â Along B C) â Along E

Review of Waves from 1201 • The one-dimensional wave equation: has a general solution of the form: where h 1 represents a wave traveling in the +x direction and h 2 represents a wave traveling in the -x direction. • A specific solution for harmonic waves traveling in the +x direction is: h l A x A = amplitude l = wavelength f = frequency v = speed k = wave number

E & B in Electromagnetic Wave • Plane Harmonic Wave: where: y x z • From general properties of waves : Þ

25 -2 The Propagation of EM Waves • All electromagnetic waves propagate through a vacuum at the same rate: • In materials, such as air and water, light slows down, but at most to about half the above speed • Using “tricks” of quantum mechanics – Can “stop” light in matter – More when we talk about modern physics !

Recall LC Circuits i The energy is conserved Q +++ C --- L LC: current oscillates i When the capacitor is fully charged: When the current is at maximum (Io): 0 t The maximum energy stored in the capacitor and in the inductor are the same:

Energy is Stored in fields • The energy density for a parallel plate capacitor: • The Electric field is given by: A ++++ ----- d Þ • The energy density u in the field is given by: • The energy density for a long solenoid: l • The inductance L is: r • Energy U: • The energy density N turns

Links between E & B • In LC, maximum energy in C and L are equal • If C and L have the same volume: u. E = u. B • E & B are related : • Units: e 0 = 8. 885× 10− 12 C 2/Nm 2 m 0 = 12. 566× 10− 7 N/A 2 • We can turn this into an energy density by dividing by the volume containing the field:

More about c • Units: e 0 = 8. 885× 10− 12 C 2/Nm 2 • m 0 = 12. 566× 10− 7 N/A 2 • [e 0 m 0 ] = (C 2/Nm 2) (N/A 2 )=(C 2/m 2) (s 2/C 2 )= s 2/ m 2 • So [e 0 m 0 ] = [1/v 2] • We have • Using values above: c=299, 276, 596 m/s ~ 3 X 108 m/s • And or

Velocity of Electromagnetic Waves • The wave equation for Ex: (derived from Maxwell’s Eqn) • Therefore, we now know the velocity of electromagnetic waves in free space: • Putting in the measured values for m 0 & e 0, we get: • This value is identical to the measured speed of light! – We identify light as an electromagnetic wave.

Recap of Today’s Topic : • Announcements: Team problems for. Thursday – Team 7: Cailin Catarina, Matthew Canapetti, Kervin Vincent – Team 8: Natalie Kasir, Adam Antunes, Quincy Alexander – Team 9: Garrett Schlegel, Joyce Nieh, Matthew Lombardo • Office hours: Monday 2: 30 -3: 30 Thursday 3: 00 -4: 00 • Homework #6: due this coming Friday • Midterm 1: – Solutions in class – Midterm sample + To-Know sheet on web already • Chapter 25: E&M waves – Production and properties