16.2 Plane electromagnetic waves  (Page 3/5)

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${E}_{y}\left(x,t\right)=f\left(\xi \right)\phantom{\rule{1.5em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\xi =x-ct.$

It is left as a mathematical exercise to show, using the chain rule for differentiation, that [link] and [link] imply

$1={\epsilon }_{0}{\mu }_{0}{c}^{2}.$

The speed of the electromagnetic wave in free space is therefore given in terms of the permeability and the permittivity of free space by

$c=\frac{1}{\sqrt{{\epsilon }_{0}{\mu }_{0}}}.$

We could just as easily have assumed an electromagnetic wave with field components ${E}_{z}\left(x,t\right)$ and ${B}_{y}\left(x,t\right)$ . The same type of analysis with [link] and [link] would also show that the speed of an electromagnetic wave is $c=1\text{/}\sqrt{{\epsilon }_{0}{\mu }_{0}}$ .

The physics of traveling electromagnetic fields was worked out by Maxwell in 1873. He showed in a more general way than our derivation that electromagnetic waves always travel in free space with a speed given by [link] . If we evaluate the speed $c=\frac{1}{\sqrt{{\epsilon }_{0}{\mu }_{0}}},$ we find that

$c=\frac{1}{\sqrt{\left(8.85\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-12}\frac{{\text{C}}^{2}}{\text{N}·{\text{m}}^{2}}\right)\left(4\text{π}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\frac{\text{T}·\text{m}}{\text{A}}\right)}}=3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s},$

which is the speed of light . Imagine the excitement that Maxwell must have felt when he discovered this equation! He had found a fundamental connection between two seemingly unrelated phenomena: electromagnetic fields and light.

Check Your Understanding The wave equation was obtained by (1) finding the E field produced by the changing B field, (2) finding the B field produced by the changing E field, and combining the two results. Which of Maxwell’s equations was the basis of step (1) and which of step (2)?

(1) Faraday’s law, (2) the Ampère-Maxwell law

So far, we have seen that the rates of change of different components of the E and B fields are related, that the electromagnetic wave is transverse, and that the wave propagates at speed c . We next show what Maxwell’s equations imply about the ratio of the E and B field magnitudes and the relative directions of the E and B fields.

We now consider solutions to [link] in the form of plane waves for the electric field:

${E}_{y}\left(x,t\right)={E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right).$

We have arbitrarily taken the wave to be traveling in the +x -direction and chosen its phase so that the maximum field strength occurs at the origin at time $t=0$ . We are justified in considering only sines and cosines in this way, and generalizing the results, because Fourier’s theorem implies we can express any wave, including even square step functions, as a superposition of sines and cosines.

At any one specific point in space, the E field oscillates sinusoidally at angular frequency $\omega$ between $+{E}_{0}$ and $\text{−}{E}_{0},$ and similarly, the B field oscillates between $+{B}_{0}$ and $\text{−}{B}_{0}.$ The amplitude of the wave is the maximum value of ${E}_{y}\left(x,t\right).$ The period of oscillation T is the time required for a complete oscillation. The frequency f is the number of complete oscillations per unit of time, and is related to the angular frequency $\omega$ by $\omega =2\pi f$ . The wavelength $\lambda$ is the distance covered by one complete cycle of the wave, and the wavenumber k is the number of wavelengths that fit into a distance of $2\text{π}$ in the units being used. These quantities are related in the same way as for a mechanical wave:

$\omega =2\pi f,\phantom{\rule{1.2em}{0ex}}f=\frac{1}{T},\phantom{\rule{1.2em}{0ex}}k=\frac{2\pi }{\lambda },\phantom{\rule{1.2em}{0ex}}\text{and}\phantom{\rule{1.2em}{0ex}}c=f\lambda =\omega \text{/}k.$

Given that the solution of ${E}_{y}$ has the form shown in [link] , we need to determine the B field that accompanies it. From [link] , the magnetic field component ${B}_{z}$ must obey

what is flux
Total number of field lines crossing the surface area
Kamru
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
prince
what is temperature change
Celine
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
Celine
process whereby the degree of hotness of a body (or medium) changes
Salim
Q=mcΔT
Salim
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
Salim
what was the temperature of the soft drink when it was removed ?
Salim
15 degree Celsius
Celine
15 degree
Celine
ok I think is just conversion
Salim
15 degree Celsius to Fahrenheit
Salim
0 degree Celsius = 32 Fahrenheit
Salim
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
Salim
I dont understand
Celine
the question said you should convert 15 degree Celsius to Fahrenheit
Salim
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
Salim
what is d final ans for Fahrenheit and Celsius
Celine
it said what is temperature change in Fahrenheit and Celsius
Celine
the 15 is already in Celsius
Salim
So the final answer for Fahrenheit is 59
Salim
what is d final ans for Fahrenheit and Celsius
Celine
what are the effects of placing a dielectric between the plates of a capacitor
increase the capacitance.
Jorge
besides increasing the capacitance, is there any?
Bundi
mechanical stiffness and small size
Jorge
why for an ideal gas internal energy is directly proportional to thermodynamics temperature?
two charged particles are 8.45cm apart. They are moved and the force on each of them is found to have tripled. How far are they now?
what is flux
Bundi
determining dimensional correctness
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
what are the elementary compositions of a cell?
poles, chemical
prabir
when a current pass through a material does the velocity varies
no.
prabir
what is spin entropy ?and disorder in ferromagnetic material
diagram of an hall effect sensor
if a magnetised wire having dipole moment M is bent in the form of arc subtending angle of 45°at centre,new magnetic moment is
prabir
is this book for preparing IIT or neet?
is it possible to increase the temperature of a gas without adding heat to it?
I'm not sure about it, but I think it's possible. If you add some form of energy to the system, it's a possibility. Also, if you change the pression or the volume of the system, you'll increase the kinetic energy of the system, increasing the gas temperature. I don't know if I'm correct.
playdoh
For example, if you get a syringe and close the tip(sealing the air inside), and start pumping the plunger, you'll notice that it starts getting hot. Again, I'm not sure if I am correct.
playdoh
you are right for example an adiabatic process changes all variables without external energy to yield a temperature change. (Search Otto cycle)
when a current pass through a material does the velocity varies
lovet
yes at adiabatic compression temperature increase
Nepal
how to draw a diagram of a triode
whate is fckg diagrame?
Arzoodan
why do we use integration?
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan