When a battery charges a parallel-plate capacitor, the battery does work separating the charges. If the battery moves a total amount of charge Q by moving electrons from the positively charged plate to the negatively charged plate, then the voltage across the capacitor is V = Q/C and the amount of work done by the battery is (1/2)CV2. The battery has converted chemical energy into electrostatic potential energy.
Where is this energy stored?
 | We can view the energy U as being stored in the separated charges, U = (1/2)Q2/C. | ||
| We can also view the energy as being stored in the electric field produced by the
separated charges, U = (1/2)CV2.
In electrostatics, viewing the energy as being stored in the separated charges or viewing it as being stored in the electric field leads to the same results. We are allowed to take either point of view.
|
Assume you connect a battery to a coil with cross sectional area A, length l, and n turns per unit length. Assume that the coil has resistance R. After some time a steady current If will flow through the coil, If = V/R, where V is the battery voltage. But just after you connect the battery, when the current begins to flow and the magnetic flux through the coil begins to change, an induced emf with magnitude e = LdI/dt will oppose the current flow. The battery has to do work against this emf. The amount of work done per unit time against the induced emf, i.e. the power supplied by the battery to overcome the induced emf, is dW/dt = P = eI = LIdI/dt, where L is the self inductance of the coil. This is the rate at which the battery converts chemical energy into magnetic energy. The total amount of energy converted is
.
| We can view this energy as being stored in the circulating current. | ||
| But we can also view the energy as being stored in the magnetic field.
In magnetostatics, viewing the energy as being stored in the circulating currents or viewing it as being stored in the magnetic field leads to the same results. We are allowed to take either point of view.
|
Electrodynamics
requires us to view the energy as being stored in the electromagnetic field. Electromagnetic (EM) waves can transport this energy across empty space. The equations of electrodynamics are Maxwell's equations.Maxwell's equations.
| (1) |  |
| (2) |  |
| (3) |  |
| (4) |  |
Electromagnetic waves are solutions to Maxwell's equations. In the last module we studied equation 2 (Faraday's law). It tells us that changing magnetic fields can produce electric fields. The circulation of the electric field around any closed loop G is proportional to the rate of change of the magnetic flux through the loop.
Let us now investigate equation 4. Ampere's law in magnetostatics becomes the Ampere-Maxwell law in electrodynamics. Magnetic fields are produced by currents, but also by changing electric fields. The circulation of the magnetic field around any closed loop G is equal to the sum of m0Ithrough_G and m0e0 = 1/c2 times the rate of change of the electric flux through the loop. Equation 4 tells us that changing electric fields can produce magnetic fields.
In regions of space free of charges and currents, we still can have electric and magnetic fields. Coulomb's law and the Biot-Savart law tell us that the static electric and magnetic fields produced by charges at rest and steady currents in other regions extend towards infinity, but their magnitude falls off as the inverse square of the distance from the charges and currents. At large distances from the charges and currents, the fields are very weak.
Static fields, however, are not the only solutions to Maxwell's equations. Electromagnetic waves are solutions to Maxwell's equations in free space, when Q and I are zero. Electromagnetic waves are changing electric and magnetic fields, carrying energy through space. Sinusoidal plane waves are one type of electromagnetic waves. Not all EM waves are sinusoidal plane waves, but all electromagnetic waves can be viewed as a linear superposition of sinusoidal plane waves traveling in arbitrary directions. A plane EM wave traveling in the x-direction is of the form
E(x,t) = Emaxcos(kx-wt+f),
B(x,t) = Bmaxcos(kx-wt+f).
For electromagnetic waves E and B are always perpendicular to each other, and perpendicular to the direction of propagation. The direction of propagation is the direction of E´B. Electromagnetic waves are transverse waves.
Maxwell's equations require that v = c = 3´108m/s for any electromagnetic wave in free space. The speed of any electromagnetic waves in free space is the speed of light c. Electromagnetic waves can have any wavelength l or frequency f as long as lf = c. When an electromagnetic wave travels through free space, Maxwell's equations require that at every instant and at any point the ratio of the electric to the magnetic field in SI units is equal to the speed of light, E/B = c.
When electromagnetic waves travel through a medium the speed of the waves in the medium is v = c/n, where n is the index of refraction of the medium. In a medium we replace the permittivity of free space e0 by the permittivity of the medium e = kee0, and we replace the permeability of free space m0 by the permeability of the medium m = kmm0. We then have
v2 = 1/(em) = c2/kekm,
or
v = c/(kekm)1/2 = c/n.
ke, km, and therefore the index of refraction n are properties of the medium. For air ke, km, and n are nearly equal to 1.
Problems:
| Determine the speed of light in water, which has a dielectric constant of
1.78.
| ||
| An electromagnetic wave in vacuum has an electric field amplitude of
220V/m. Calculate the amplitude of the corresponding magnetic field.
|
For information about the electromagnetic spectrum see Physics 135, Electromagnetic waves
Like all EM waves, light transports energy across space. The intensity (energy per unit area and unit time) is proportional to the square of the amplitude of the electric field of the light wave. This energy, however, arrives at a receiver not continuously but in discrete units called photons. The energy transported by an electromagnetic wave is not continuously distributed over the wavefront. It is transported in discrete packages. In addition to its wave properties, light also has particle nature. Photons are the particles of light.
| Photons always move with the speed of light. | ||
| Photons are electrically neutral. | ||
| Photons have no mass, but they have energy E =
hf = hc/l.
Here
h = 6.626´10-34Js is a universal constant called Planck's constant.
The energy of each photon is
inversely proportional to the wavelength of the associated EM wave. The shorter the
wavelength, the more energetic is the photon, the longer the wavelength, the less
energetic is the photon.
| ||
| Photons can be created and destroyed. When a source emits EM waves, photons are created. When photons encounter matter, they may be absorbed and transfer their energy to the atoms and molecules. Creation and destruction of photons must conserve energy and momentum. The momentum of a photon is p = hf/c. |
Problem:
| What is the energy of a photon of blue light (l
= 450nm) and of
a photon of red light (l =
700nm) in units of eV =
1.6´10-19J?
|
The wave nature and the particle nature of light are complementary properties. Experiments probing the propagation of light through a medium and around obstacles reveal the wave nature of light. Experiments probing energy and momentum conservation when light interacts with atoms and molecules reveal the particle nature of light. No single experiment has ever revealed both the wave and particle nature simultaneously.
In 1905, Einstein used the discrete nature of light to explain the photoelectric effect. To demonstrate this effect light is shone on a metal surface. If the frequency of the light is higher than the cutoff frequency fc, then electrons are released. No photoelectric electrons are emitted if the frequency of the light falls below this cutoff frequency fc. For many metal surfaces the frequency of blue light is greater than fc and the frequency of red light is less than fc. If red light is shone on the surface, no electrons are emitted, no matter what the intensity of the light. If blue light is shone on the surface, electrons are emitted. The number of emitted electrons depends on the intensity of the light. But even if the intensity is reduced to a very low value, electrons are still emitted, albeit at a very low rate.
The photoelectric effect cannot be understood within the wave picture of light. To eject an electron from a metal surface a certain amount of energy f, called the work function of the metal, must be supplied to this electron. In the wave picture the energy of the light beam does not depend on the frequency, but only on the intensity. Einstein explained the photoelectric effect by postulating that an electron can only receive the large amount of energy necessary to escape the metal from the EM wave by absorbing a single photon. If this photon has enough energy, the electron is freed. Excess energy appears as kinetic energy of the electron. The kinetic energy of the electron is given by E = hf-f. If the photon does not have enough energy, then the electron cannot escape the metal.
Problems:
| Molybdenum has a work function of 4.2eV. Find the cutoff frequency and cutoff wavelength
for the photoelectric effect.
| ||||||
| Electrons are ejected from a metal surface with speeds ranging up to 4.6´105m/s when light with a wavelength of 625nm is used. (a)
What is the work function of the surface?
|
Links:
| What is the photoelectric effect? |
| The photoelectric effect applet |
| A charged particle produces an electric field. This electric field exerts a force on other charged particles. Positive charges accelerate in the direction of the field and negative charges accelerate in a direction opposite to the direction of the field. |
| A moving charged particle produces a magnetic field. This magnetic field exerts a force on other moving charges. The force on these charges is always perpendicular to the direction of their velocity and therefore only changes the direction of the velocity, not the speed. |
| An accelerating charged particle produces an electromagnetic wave. Electromagnetic waves are electric and magnetic fields traveling through space with the speed of light c. A charged particle oscillating about an equilibrium position is an accelerating charged particle. If its frequency of oscillation is f, then it produces an electromagnetic wave with frequency f. The wavelength l of this wave is given by l = c/f. Electromagnetic waves transport energy through space. This energy can be delivered to charged particles a large distance away from the source. |
Assume a charge q is accelerating. It therefore produces electromagnetic radiation. At some position r in space and at some time t, the electric field of the electromagnetic wave produced by the accelerating charge is given by
where
.
Let us analyze this expression. The electric field is proportional to the charge q. The bigger the accelerating charge, the bigger is the field. It decreases as the inverse of the distance r'', which is the distance between the accelerating charge and the position where the field is observed. But it is not the distance at the time the field is observed, but the distance at some earlier time, called the retarded time, when the radiation field was produced. Since electromagnetic waves travel with speed c, it takes them a time interval Dt = Dr/c to travel a distance Dr. The electric field is also proportional to the acceleration of the charge. The larger the acceleration, the larger is the field. The directional aspects are given by E(r,t) µ a^. The direction of the electric field is perpendicular to the line of sight between r and the retarded position of the charge and its magnitude is proportional to the component of the acceleration perpendicular to this line of sight. The figure below illustrates that point. The electric field is zero along a line of sight in the direction of the acceleration, largest along a line of sight perpendicular to the direction of the acceleration, and always perpendicular to the line of sight.
The magnetic field is perpendicular to the electric field and to the direction of propagation r'', and its magnitude is B = E/c. This can be written as
.
The power radiated away by the accelerating charge is
,
with
.
This is called the Lamor formula.
AM radio waves have a frequency between 550 kHz and 1600 kHz and FM wave have a frequency between 88 MHz and 108 MHz. To produce radio waves, charges have to oscillate with frequencies in this range. Usually the charges oscillate in a resonance or tank circuit. A tank circuit is shown below. It consists of a capacitor with capacitance C and a coil with self-inductance L.
Assume that initially one of the capacitor plates is positively charged and the other is negatively charged. (You can charge the capacitor by connecting the terminals of a battery to the plates. The amount of charge Q on each plate is proportional to the battery voltage V, Q = CV.) Separated charges produce a net electric field, and electrons will accelerate everywhere in the circuit, and a current will start flowing. The moving charges produce a magnetic field. (The magnetic field in the coil is proportional to its self inductance L times the current flowing through the coil.)
As the magnetic field in the coil changes, it produces an electric field opposing the current flow (Lenz's rule) and thus slowing down the rate at which the capacitor will discharge. When the capacitor is completely discharged, the electrons still have kinetic energy. This kinetic energy is converted into potential energy as they start piling onto the other plate producing an electric field opposing their motion and decreasing the current. As the current decreases, the magnetic field in the coil decreases. The collapsing magnetic field now produces an electric field, which slows down the rate at which the current falls to zero. When the current reaches zero the charges on the capacitor plates will be reversed, and the process will start over again with the current flowing in the opposite direction.
The circuit has a natural oscillation frequency f, which depends on C and L.
.
Unless energy is constantly pumped into the circuit by a power supply, the oscillations will eventually die out in a real circuit. Some of the energy in the circuit will go into resistive heating, and some will be radiated away by the accelerating charges. But if the power supply excites the circuit with the natural frequency f, it only has to re-supply the lost energy, and not the total energy stored in the circuit per cycle. The amount of energy radiated away by an oscillating charge per second is proportional to the magnitude of the charge and to the fourth power of its frequency. The higher the frequency, the greater is the amount of energy lost to radiation.
When an electromagnetic wave passes an antenna connected to a tank circuit, its electric field will cause the electrons in the antenna and in the circuit to oscillate. If the frequency of the electromagnetic wave matches the natural frequency of the tank circuit, then it can efficiently transfer energy to the circuit and increase the amplitude of the oscillations. The oscillations of the circuit then follow the oscillations of the wave. If the frequency of the wave does not match the frequency of the circuit, then the small fields produce at most some random jiggling of the electrons (noise). If the capacitance of the tank circuit can be adjusted, then the tank circuit becomes a tunable radio receiver.
Most radio waves are emitted by charges oscillating in antennas. The direction of the acceleration of the charges is along the antenna. The direction of the electric field E of the electromagnetic radiation emitted by the antenna lies in a plane that contains the antenna and the line of sight to the receiver, and is perpendicular to the line of sight. The wave is polarized.
The electric field is strongest and the intensity highest in the directions perpendicular to the antenna and goes to zero in the direction along the antenna.
To carry information the electromagnetic wave must be modulated. The information carried by a radio wave is sound. The amplitude of an AM (amplitude modulated) radio wave represents the pressure variations, which make up the sound. The frequency of FM (frequency modulated) radio waves can be shifted slightly from their nominal carrier frequency. The amount of shift is proportional to the variations in the pressure, which make up the sound.
Links:
| Amplitude Modulation |
| Frequency Modulation |
| How the radio spectrum works |
A cathode-ray tube (CRT) is a specialized vacuum tube in which images are produced when an electron beam strikes a phosphorescent surface. A cathode-ray tube consists of several basic components, as illustrated below.
The electron gun generates a narrow beam of electrons. The anodes accelerate and focus the electrons. Deflecting coils produce a magnetic field that allows for adjustment of the direction of the electron beam. There are two sets of deflecting coils: horizontal and vertical. (In the illustration, only one set of coils is shown for simplicity.) T he electron beam produces a tiny, bright visible spot when it strikes the phosphor-coated screen.
In a basic black and white TV, a single electron beam is caused to race across the screen from left to right and from top to bottom as viewed from the front of the CRT, in a sequence of horizontal lines called the raster. After each line is written, and the beam returns back to the left, the signal is blanked. When the signal reached the bottom it is blanked until it returns to the top to write the next line. The scan is interlaced to reduce flicker, it scans twice per photographed frame. The first scan includes only the odd lines, the next scan includes only the even lines. NTSC systems have a field rate of 59.94 Hz and a frame rate of ~30Hz. To produce an image on the screen a signal controls the intensity of the electron beam.
Conventional NTSC has 525 vertical lines. However lines number 248 to 263 and 511 to 525 are typically blanked to provide time for the beam to return to the upper left hand corner for the next scan. The electron beam is analog modulated across the horizontal line. The modulation then translates into intensity changes in the electron beam and thus gray scale levels on the picture screen.
Color television was developed in such a way that it remained compatible with black and white television. Black-and-white TVs continue to be able to receive a valid TV signal.
The "additive color mixing" scheme is used to display color.
A standard TV has three dots (dot triad) at each location on the screen; red, green and blue. There is a corresponding electron gun for each color that emits an electron beam of varying intensity - this corresponds to color brightness. To ensure that the electrons from each gun strike the corresponding phosphor, a "shadow mask" is used.
Because the three electron beams arrive at slightly different angles (from the three separate electron guns), it is possible to construct and align the shadow mask such that the electron beam from one gun will strike the correct phosphor dot, but the other two phosphors will be in shadow. This way, the intensity of red, green and blue can be separately controlled at each dot triad location. After a beam leaves a phosphor dot, the phosphor continues to glow briefly, this condition is called persistence. For an image to remain stable, the phosphors must be reactivated by repeated scans of the electron beams.
Link:
| TV and laptop screens
|
Polarization is a phenomenon peculiar to transverse waves. Longitudinal waves such as sound cannot be polarized. Light and other electromagnetic waves are transverse waves made up of mutually perpendicular, fluctuating electric and magnetic fields. In the diagram below an EM wave is propagating in the x-direction, the electric field oscillates in the xy plane, and the magnetic field oscillates in the xz plane. A line traces out the electric field vector as the wave propagates.
An unpolarized electromagnetic wave traveling in the x-direction is a superposition of many waves. For each of these waves the electric field vector is perpendicular to the x-axis, but the angle it makes with the y-axis is different for different waves. For a polarized electromagnetic wave traveling in the x-direction, the angle the electric field makes with the y-axis is unique. A polarizer is a material that passes only EM waves for which the electric field vector is confined to a single plane that contains to the direction of motion.
An ideal polarizer passes the components of the electric field vectors that are parallel to its transmission axis. If E0 is the incident field vector and the angle between E0 and the transmission axis is q, then the magnitude of transmitted field vector is E0cosq and its direction is the direction of the transmission axis. The intensity I of an electromagnetic wave is proportional to the square of the magnitude of the electric field vector. We therefore have
Itransmitted =I 0cos2q.
If q = 90o the transmitted intensity is zero.
The TV signal controls the intensity of the electron beam. The luminance or combined brightness of the three guns is transmitted via the amplitude variation of the carrier signal (55.25 MHz for channel 2 for example). Amplitude and frequency modulations of a carrier frequency produce sidebands, and so the TV signal is spread over a range of frequencies or bandwidth. The luminance signal also contains sync signals (spikes), which are interpreted by the circuitry in the TV to steer the electron beam to begin a new line.
The chrominance or color information is transmitted by modulating a wave with a subcarrier frequency 3.85 MHz above the carrier frequency. A rather complicated scheme involving both the amplitude and the phase is used. The circuitry in the TV has to decode these signals before actually controlling the relative brightness the three separate electron beams. The sound is transmitted by FM modulation of a separate carrier wave.
Additional Links:
| Microwave ovens |
| How a microwave oven works |
| The Magnetron Tube |
Please complete assignment 18 now. For the User ID use your registered password.
You can submit the assignment up to three times. Each time the computer will tell you your score.
.