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Eðlisfræði 2, vor 2007
32. Electromagnetic Waves
Assignment is due at 2:00am on Wednesday, March 28, 2007
Credit for problems submitted late will decrease to 0% after the deadline has passed.
The wrong answer penalty is 2% per part. Multiple choice questions are penalized as described in the online help.
The unopened hint bonus is 2% per part.
You are allowed 4 attempts per answer.
Travelling E-M Waves
Traveling Electromagnetic Wave
Learning Goal: To understand the formula representing a traveling electromagnetic wave.
Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves comprise combinations of electric and magnetic fields
that are mutually compatible in the sense that the changes in one generate the other.
The simplest form of a traveling electromagnetic wave is a plane wave. For a wave traveling in the x direction whose electric field is in the y direction, the electric and magnetic fields are given by
,
.
This wave is linearly polarized in the y direction.
Part A
In these formulas, it is useful to understand which variables are parameters that specify the nature of the wave. The variables
Hint A.1
and
are the __________ of the electric and magnetic fields.
What are parameters?
Hint not displayed
Choose the best answer to fill in the blank.
ANSWER:
maxima
amplitudes
wavelengths
velocities
Part B
The variable
is called the __________ of the wave.
Choose the best answer to fill in the blank.
ANSWER:
velocity
angular frequency
wavelength
Part C
The variable
is called the __________ of the wave.
Choose the best answer to fill in the blank.
ANSWER:
wavenumber
wavelength
velocity
frequency
Part D
What is the mathematical expression for the electric field at the point
at time ?
ANSWER:
Part E
For a given wave, what are the physical variables to which the wave responds?
Hint E.1
What are independent variables?
Hint not displayed
ANSWER:
only
only
only
only
and
and
and
and
This is a plane wave; that is, it extends throughout all space. Therefore it exists for any values of the variables
and
and can be considered a function of
, , , and . Being an infinite
plane wave, however, it is independent of these variables. So whether they are considered independent variables is a question of semantics.
When you appreciate this you will understand the conundrum facing the young Einstein. If he traveled along with this wave (i.e., at the speed of light ), he would see constant electric and
magnetic fields extending over a large region of space with no time variation. He would not see any currents or charge, and so he could not see how these fields could satisfy the standard
electromagnetic equations for the production of fields.
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Part F
What is the wavelength
Hint F.1
of the wave described in the problem introduction?
Finding the wavelength
The wave described in the introduction is sinusoidal. If we let
, then the spatial dependence of the wave is given by
direction within which the wave repeats itself. Mathematically, we require
. To find
. The wavelength
is defined to be the length in the x
, recall that the sine function repeats itself when its argument changes by
:
.
Express the wavelength in terms of the other given variables and constants like
ANSWER:
.
=
Part G
What is the period
of the wave described in the problem introduction?
Express the period of this wave in terms of
ANSWER:
and any constants.
=
Part H
What is the velocity
Hint H.1
of the wave described in the problem introduction?
How to find
Hint not displayed
Express the velocity in terms of quantities given in the introduction (such as
ANSWER:
and ) and any useful constants.
=
If this electromagnetic wave were traveling in a vacuum its velocity would be equivalent to , the vacuum speed of light.
Solving M's Eqns to Find c
Triangle Electromagnetic Wave
Learning Goal: To show how a propagating triangle electromagnetic wave can satisfy Maxwell's equations if the wave travels at speed c.
Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves consist of mutually compatible combinations of electric
and magnetic fields ("mutually compatible" in the sense that changes in the electric field generate the magnetic field, and vice versa).
The simplest form for a traveling electromagnetic wave is a plane wave. One particularly simple form for a plane wave is known as a "triangle wave," in which the electric and magnetic fields are
linear in position and time (rather than sinusoidal). In this problem we will investigate a triangle wave traveling in the x direction whose electric field is in the y direction. This wave is linearly
polarized along the y axis; in other words, the electric field is always directed along the y axis. Its electric and magnetic fields are given by the following expressions:
and
where
,
, and
are constants. The constant , which has dimensions of length, is introduced so that the constants
This wave is pictured in the figure at time
,
and
have dimensions of electric and magnetic field respectively.
.
Note that we have only drawn the field vectors along the x axis. In fact, this idealized wave fills all space, but the field vectors only
vary in the x direction.
We expect this wave to satisfy Maxwell's equations. For it to do so, we will find that the following must be true:
1 . The amplitude of the electric field must be directly proportional to the amplitude of the magnetic field.
2 . The wave must travel at a particular velocity (namely, the speed of light).
Part A
What is the propagation velocity
Part A.1
of the electromagnetic wave whose electric and magnetic fields are given by the expressions in the introduction?
Phase velocity
Part not displayed
Express
in terms of
and the unit vectors
, , and . The answer will not involve ; we have not yet shown that this wave travels at the speed of light.
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ANSWER:
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=
In the next few parts, we will use Faraday's law of induction to find a relationship between
and
.
Faraday's law relates the line integral of the electric field around a closed loop to the rate of change in magnetic flux through this loop:
.
Part B
To use Faraday's law for this problem, you will need to constuct a suitable loop, around which you will integrate the electric field. In which plane should the loop lie to get a nonzero electric field
line integral and a nonzero magnetic flux?
ANSWER:
the xy plane
the yz plane
the zx plane
Part C
Consider the loop
shown in the figure. It is a square loop with sides of length
, with one corner at the origin and the opposite corner at the coordinates
. What is the value of the line integral of the electric field around loop
Part C.1
,
. Recall that
at arbitrary time ?
Integrating along segments 1 and 2
Part not displayed
Part C.2
Integrating along segments 3 and 4
Part not displayed
Hint C.3
Integrating around the entire loop
Hint not displayed
Express the line integral in terms of
ANSWER:
,
, , , and/or .
=
Part D
Recall that
. Find the value of the magnetic flux through the surface
Hint D.1
in the xy plane that is bounded by the loop
, at arbitrary time .
Simplifying the integrand
Hint not displayed
Part D.2
Evaluating the integral
Part not displayed
Express the magnetic flux in terms of
ANSWER:
,
, , , and/or .
=
Part E
Now use Faraday's law to establish a relationship between
Part E.1
and
.
Using Faraday's law
Part not displayed
Express
in terms of
ANSWER:
and other quantities given in the introduction.
=
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If the electric and magnetic fields given in the introduction are to be self-consistent, they must obey all of Maxwell's equations, including the Ampère-Maxwell law. In these last few parts (again,
most of which are hidden) we will use the Ampère-Maxwell law to show that self-consistency requires the electromagnetic wave described in the introduction to propagate at the speed of light.
The Ampère-Maxwell law relates the line integral of the magnetic field around a closed loop to the rate of change in electric flux through this loop:
.
In this problem, the current
is zero. (For
to be nonzero, we would need charged particles moving around. In this problem, there are no charged particles present. We assume that the
electromagnetic wave is propagating through a vacuum.)
Part F
To use the Ampère-Maxwell law you will once again need to construct a suitable loop, but this time you will integrate the magnetic field around the loop. In which plane should the loop lie to get
a nonzero magnetic field line integral and hence nonzero electric flux?
ANSWER:
the xy plane
the yz plane
the zx plane
Part G
Use the Ampère-Maxwell law to find a new relationship between
Hint G.1
and
.
How to approach the problem
Hint not displayed
Part G.2
Find an expression for the left-hand side of the equation
Part not displayed
Part G.3
Find an expression for the right-hand side of the equation
Part not displayed
Part G.4
Use the Ampère-Maxwell law
Part not displayed
Express
in terms of
,
ANSWER:
,
, and other quantities given in the introduction.
=
Part H
Finally we are ready to show that the electric and magnetic fields given in the introduction describe an electromagnetic wave propagating at the speed of light. If the electric and magnetic fields
are to be self-consistent, they must obey all of Maxwell's equations. Using one of Maxwell's equations, Faraday's law, we found a certain relationship between
and
. You derived this in
Part E. Using another of Maxwell's equations, the Ampère-Maxwell law, we found what appears to be a different relationship between
E and I are to agree, what does this imply that the speed of propagation
Express
in terms of only
ANSWER:
and
and
. You derived this in Part I. If the results of Parts
must be?
.
=
You have just worked through the details of one of the great triumphs of physics: Maxwell's equations predict a form of traveling wave consisting of a matched pair of electric and magnetic
fields moving at a very high velocity
. We can measure
and independently in the laboratory, and these experimentally determined values lead to a speed of
, the speed of light . After thousands of years of speculation about the nature of light, Maxwell had developed a plausible and quantitatively testable theory about it.
Faraday had a hunch that light and magnetism were related, as demonstrated by the Faraday effect. (Glass, put in a large magnetic field, will rotate the plane of polarization of light that
passes through it.) Now Maxwell had predicted an electromagnetic wave with the following properties:
1 . It was transverse, with two possible polarizations (which agreed with an already known characteristic of light).
2 . It had an extraordinarily high velocity (relative to waves in air or on strings) that agreed with the experimentally determined value for the speed of light.
Any doubt that light waves were in fact electromagnetic waves vanished as various optical phenomena (such as the behavior of electromagnetic waves at glass surfaces) were predicted and
found to agree with the behavior of light. This theory showed that lower frequency waves could be created and detected by their interactions with currents in wires (later called antennas) and
paved the way to the creation and detection of radio waves.
Poynting Vector and Power in E-M Waves
Poynting Flux and Power Dissipation in a Resistor
When a steady current flows through a resistor, the resistor heats up. We say that "electrical energy is dissipated" by the resistor, that is, converted into heat. But if energy is dissipated, where did it
come from? Did it come from the voltage source through the wires?
This problem will show you an alternative way to think about the flow of energy and will introduce a picture in which the energy flows in many unexpected places--but not through the wires!
We will calculate the Poynting flux, the flow of electromagnetic energy, across the surface of the resistor. The Poynting flux, or Poynting vector
and is related to the electric field vector
and the magnetic field vector
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, has units of energy per unit area per unit time
by the equation
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,
where
is the permeability of free space.
Consider a cylindrical resistor of radius
, length , and resistance
with a steady current
flowing along the axis of the cylinder.
Part A
Which of the following is the most accurate qualitative description of the the magnetic field vector
ANSWER:
inside the cylindrical resistor?
Answer not displayed
Part B
Part not displayed
Part C
What can you say about the electric field vector
ANSWER:
inside the resistor?
Answer not displayed
Part D
Part not displayed
Part E
In what direction does the Poynting vector
Hint E.1
point?
Cross products in cylindrical coordinates
Hint not displayed
ANSWER:
Answer not displayed
Part F
Part not displayed
Poynting Flux
An electromagnetic wave is traveling through vacuum. Its electric field vector is given by
,
where
is the unit vector in the y direction.
Part A
If
is the amplitude of the magnetic field vector, find the complete expression for the magnetic field vector
Hint A.1
Relative orientation of
and
of the wave.
for a wave in vacuum
Hint not displayed
Hint A.2
Orientation of
and
relative to the direction of propagation
Hint not displayed
Part A.3
Determine the direction of propagation of the wave
Part not displayed
Hint A.4
Phase relationship between
and
Hint not displayed
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ANSWER:
Part B
What is the Poynting vector
Hint B.1
, that is, the power per unit area associated with the electromagnetic wave described in the problem introduction?
Definition of the Poynting vector
Hint not displayed
Give your answer in terms of some or all of the variables
ANSWER:
,
, ,
,
, , and
. Specify the direction of the Poynting vector using the unit vectors
, , and
as appropriate.
=
Energy in Electromagnetic Waves
Electromagnetic waves transport energy. This problem shows you which parts of the energy are stored in the electric and magnetic fields, respectively, and also makes a useful connection between
the energy density of a plane electromagnetic wave and the Poynting vector.
In this problem, we explore the properties of a plane electromagnetic wave traveling at the speed of light
along the x axis through vacuum. Its electric and magnetic field vectors are as follows:
.
Throughout, use these variables (
,
,
,
, ,
, and
) in your answers. You will also need the permittivity of free space
and the permeability of free space
.
Note: To indicate the square of a trigonometric function in your answer, use the notation sin(x)^2 NOT sin^2(x).
Part A
What is the instantaneous energy density
Hint A.1
in the electric field of the wave?
Energy density in an electric field
Hint not displayed
Give your answer in terms of some or all of the variables in
ANSWER:
.
=
Part B
What is the instantaneous energy density
Hint B.1
in the magnetic field of the wave?
Energy density in a magnetic field
Hint not displayed
Give your answer in terms of some or all of the variables in
ANSWER:
.
=
Part C
What is the average energy density
Hint C.1
in the electric field of the wave?
Average value of
Hint not displayed
Give your answer in terms of
ANSWER:
and
.
=
Part D
What is the average energy density
in the magnetic field of the wave?
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Hint D.1
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Average value of
Hint not displayed
Give your answer in terms of
ANSWER:
and
.
=
Part E
From the previous results, derive an expression for
Hint E.1
Relationship among
,
, the average energy density in the whole wave.
, and
Hint not displayed
Hint E.2
Relationship between
and
for electromagnetic waves in vacuum
Hint not displayed
Hint E.3
Relationship among ,
and
for electromagnetic waves in vacuum
Hint not displayed
Express the average energy density in terms of
ANSWER:
and
only.
=
Part F
The Poynting vector
gives the energy flux per unit area of electromagnetic waves. It is defined by the relation
.
Calculate the time-averaged Poynting vector
Hint F.1
Relationship between
of the wave considered in this problem.
and
for electromagnetic waves in vacuum
Hint not displayed
Hint F.2
Relationship among ,
and
for electromagnetic waves in vacuum
Hint not displayed
Give your answer in terms of
ANSWER:
,
and
and unit vectors
, , and/or . Do not use
or
.
=
If you compare this expression for the time-averaged Poynting flux to the one obtained for the overall energy density, you find the simple relation
.
Thus, the energy density of the electromagnetic field times the speed at which it moves gives the energy flux, which is a logical result.
Radiation Pressure
Radiation Pressure
A communications satellite orbiting the earth has solar panels that completely absorb all sunlight incident upon them. The total area
of the panels is
.
Part A
The intensity of the sun's radiation incident upon the earth is about
total solar power
Hint A.1
. Suppose this is the value for the intensity of sunlight incident upon the satellite's solar panels. What is the
absorbed by the panels?
Definition of intensity
Hint not displayed
Express your answer numerically in kilowatts to two significant figures.
ANSWER:
= Answer not displayed kW
Part B
What is the total force
on the panels exerted by radiation pressure from the sunlight?
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Hint B.1
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Time derivative of a kinetic energy in relation to momentum
Hint not displayed
Part B.2
Working out the power incident upon the panels
Part not displayed
Hint B.3
Getting the units right
Hint not displayed
Express the total force numerically, to two significant figures, in units of newtons.
ANSWER:
= Answer not displayed N
Solar Sail
A solar sail allows a spacecraft to use radiation pressure, instead of rockets, for propulsion (similar to the way wind propels a sailboat). The sails of such spacecraft are usually made out of a large
reflecting panel. The size of each panel is maximized to allow the largest possible flux of incident photons, leading to the largest possible total momentum transfer from the incident radiation.
Because the surface is reflective, the momentum transferred by the photons is twice what they carry. For such spacecraft to work, the force from the radiation pressure exerted by the photons must
be greater than the gravitational attraction from the star providing the photons. The critical parameter turns out to be the mass per unit area of the sail.
To solve this problem you will need to know the following:
mass of the sun:
,
intensity of sunlight as a function of the distance
from the sun:
,
and
gravitational constant:
.
Part A
Suppose that a perfectly reflecting circular mirror is initially at rest a distance
away from the sun and is oriented so that the solar radiation is incident upon, and perpendicular to, the plane of
the mirror. What is the critical value of mass/area for which the radiation pressure exactly cancels out the force due to gravity?
Part A.1
Find the force due to radiation
Part not displayed
Part A.2
Find the force due to gravity
Part not displayed
Hint A.3
Solving for mass/area
Hint not displayed
Express your answer numerically, to two significant figures, in units of kilograms per meter squared.
ANSWER:
mass/area = 1.60×10 −3
When choosing the material for a solar sail, density, strength, and reflectivity are the principal concerns. Given a representative thickness of the sail of 1
sufficiently low density and high strength are carbon fibers. These have a density of 1.60
, the only current material with a
, roughly one fifth that of iron.
The Electromagnetic Spectrum
Electromagnetic radiation is more common than you think. Radio and TV stations emit radio waves when they broadcast their programs; microwaves cook your food in a microwave oven; dentists
use X rays to check your teeth. Even though they have different names and different applications, these types of radiation are really all the same thing: electromagnetic (EM) waves, that is, energy
that travels in the form of oscillating electric and magnetic fields.
Consider the following:
radio waves emitted by a weather radar system to detect raindrops and ice crystals in the atmosphere to study weather patterns;
microwaves used in communication satellite transmissions;
infrared waves that are perceived as heat when you turn on a burner on an electric stove;
the multicolor light in a rainbow;
the ultraviolet solar radiation that reaches the surface of the earth and causes unprotected skin to burn; and
X rays used in medicine for diagnostic imaging.
Part A
Which of the following statements correctly describe the various forms of EM radiation listed above?
A.
B.
C.
D.
E.
They have different wavelengths.
They have different frequencies.
They propagate at different speeds through a vacuum depending on their frequency.
They propagate at different speeds through nonvacuum media depending on both their frequency and the material in which they travel.
They require different media to propagate.
Hint A.1
The electromagnetic spectrum
Hint not displayed
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Hint A.2
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Frequency and wavelength of an EM wave
Hint not displayed
Enter the letters of all the correct options in alphabetical order. For instance, if you thought that A, B, and D were correct, then you would enter ABD.
ANSWER:
ABD
The frequency and wavelength of EM waves can vary over a wide range of values. Scientists refer to the full range of frequencies that EM radiation can have as the electromagnetic spectrum.
Electromagnetic waves are used extensively in modern technology. Many devices are built to emit and/or receive EM waves at a very specific frequency, or within a narrow band of frequencies.
Here are some examples followed by their frequencies of operation:
garage door openers: 40.0
,
standard cordless phones: 40.0 to 50.0
baby monitors: 49.0
FM radio stations: 88.0 to 108
cell phones: 800 to 900
,
,
,
,
Global Positioning System: 1227 to 1575
microwave ovens: 2450
,
,
wireless Internet technology: 2.4 to 2.6
.
Part B
Which of the following statements correctly describe the various applications listed above?
A.
B.
C.
D.
E.
F.
All these technologies use radio waves, including low-frequency microwaves.
All these technologies use radio waves, including high-frequency microwaves.
All these technologies use a combination of infrared waves and high-frequency microwaves.
Microwave ovens emit in the same frequency band as some wireless Internet devices.
The radiation emitted by wireless Internet devices has the shortest wavelength of all the technologies listed above.
All these technologies emit waves with a wavelength in the range 0.10 to 10.0 .
G . All the technologies emit waves with a wavelength in the range 0.01 to 10.0
Hint B.1
.
Frequency and wavelength of an EM wave
Hint not displayed
Hint B.2
Hertz, megahertz, and gigahertz
Hint not displayed
Hint B.3
Meters and kilometers
Hint not displayed
Enter the letters of all the correct options in alphabetical order. For instance, if you thought that A, B, and D were correct, then you would enter ABD.
ANSWER:
ADEF
The frequency band used in wireless technology is strictly regulated by government agencies to avoid undesired interference effects. In the United States, the Federal Communications
Commission (FCC) is responsible for assigning specific radio frequency bands to different wireless communication systems.
Despite their extensive applications in communication systems, radio waves are not the only form of EM waves present in our atmosphere. Another form of EM radiation plays an even more
important role in our life (and the life of our planet): sunlight.
The sun emits over a wide range of frequencies; however, the fraction of its radiation that reaches the earth's surface is mostly in the visible spectrum. (Note that about 35% of the radiation
coming from the sun is absorbed directly by the atmosphere before even reaching the earth's surface.) The earth, then, absorbs this radiation and reemits it as infrared waves.
Part C
Based on this information, which of the following statements is correct?
A.
B.
C.
D.
The earth absorbs visible light and emits radiation with a shorter wavelength.
The earth absorbs visible light and emits radiation with a longer wavelength.
The earth absorbs visible light and emits radiation with a lower frequency.
The earth absorbs visible light and emits radiation with a higher frequency.
Hint C.1
Relation between frequency and wavelength
Hint not displayed
Enter the letters of the correct options in alphabetical order. For instance, if you thought that A, B, and D were correct, then you would enter ABD.
ANSWER:
BC
Even though our atmosphere absorbs a very small amount of visible light, it strongly reflects and absorbs infrared waves. Therefore the radiation emitted by the earth does not leave the
atmosphere. Instead, it is reflected back into it, contributing to a warming effect known as the greenhouse effect.
Part D
A large fraction of the ultraviolet (UV) radiation coming from the sun is absorbed by the atmosphere. The main UV absorber in our atmosphere is ozone,
radiation with frequencies around 9.38×10 14
. What is the wavelength
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. In particular, ozone absorbs
of the radiation absorbed by ozone?
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Hint D.1
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Frequency and wavelength of an EM wave
Hint not displayed
Hint D.2
Meters and nanometers
Hint not displayed
Express your answer in nanometers.
ANSWER:
= 320
Electromagnetic Waves and Human Vision
The photoreceptors in the human eye, called rods and cones, have different sensitivities to different wavelengths of electromagnetic waves. (Notice that the y axis in the figure is a logarithmic
scale.)
The rods, which number over 100 million, are not senstive to color. In other words, they note differences in shades of grey (from black
to white) and are responsible for a person's ability to see in dim light. Cones, which number around 6 million, are responsible for color
vision. Cones come in three different kinds: 64 of cones are sensitive to long wavelengths of visible light (toward the red end of the
spectrum), 32
are sensitive to medium wavelengths, and the remaining 2
are sensitive to short wavelengths (toward the blue end of
the spectrum). Colors are differentiated on the basis of the extent to which visible light stimulates each kind of cone.
Part A
Do rods have their peak sensitivity at a higher or lower frequency than cones?
Hint A.1
Relationship between wavelength and frequency
Hint not displayed
ANSWER:
higher
lower
Part B
Do rods and cones have similar sensitivities near the red or near the violet edge of the visible spectrum?
Hint B.1
Visible light
Hint not displayed
ANSWER:
red
violet
Part C
Is it easier to detect a dim red source or a dim violet source of light?
Hint C.1
Which curve to use
Hint not displayed
ANSWER:
red
violet
Part D
At 500
, which of the following statements is true?
Hint D.1
Logarithmic scales
Hint not displayed
ANSWER:
Rods are about 1000 times more sensitive than cones. Rods are about 3 times more sensitive than cones.
about 3 times more sensitive than rods. Cones are about 1000 times more sensitive than rods.
Rods and cones are about equally sensitive.
Cones are
Part E
Since rods are about 1000 times more sensitive than cones (at 500
), they should be able to detect smaller values of the electric field. Assuming rods and cones are sensitive to the average
energy density of an electromagnetic wave, which of the following statements is correct?
Hint E.1
Average energy density of an electromagnetic wave
Hint not displayed
Hint E.2
Relating energy density to electric field
Hint not displayed
ANSWER:
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ANSWER:
04/19/2007 05:03 PM
Rods are able to detect electric fields 1000 times smaller than the fields detectable by cones.
the fields detectable by cones.
Rods are able to detect electric fields
Rods are able to detect electric fields
times smaller than the fields detectable by cones.
times smaller than
Rods are able to detect electric
fields 3 times smaller than the fields detectable by cones.
Summary
7 of 9 problems complete (79.89% avg. score)
35.95 of 35 points
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