Electromagnetic Waves as Sinusoidal Waves

Concept: Electromagnetic Waves as Sinusoidal Waves

Video Transcript

Hey guys in this video we're going to talk about electromagnetic waves mathematically described as sinusoidal waves, let's get to it. Now as we know a common electromagnetic wave can be represented as a sinusoidal wave or sorry as an electric and magnetic field both of which are sinusoidal the image above me was the image that we saw before and this is the most common image for an electromagnetic wave we have in this case an electric field pointing in the X direction right we have an electrical in the X direction we have a magnetic field in the Y direction and we have propagation in the Z direction this makes our wave as we know a transverse wave because indeed the direction of propagation is perpendicular to both directions of oscillation. The direction of oscillation for the electric field and the direction of oscillation for the magnetic field so if we want to describe this mathematically we would write them as sine functions for example the electric field there would be some maximum electric field which is that amplitude of oscillation times sine of K X minus omega T. Omega we already know because we've seen this in oscillating functions before is the angular frequency of oscillation but K is something new it's called the wave number and I'll explain what it is and one second. Now to make this a vector we have the say in what direction the electric field is oscillating in our case it's oscillating in the X direction so I'll give it an I hat the magnetic field is going to be described the exact same way because if you look at the waves the oscillations in the above picture their identical except that the magnetic field is 90 degrees from the electric field so this is going to be some maximum magnetic field B max times sine of once again K, X minus omega T and I need to give it a direction and I'll say since its moving in the Y direction sorry since it's oscillating in the Y direction. These are Js by the way I made a little mistake here those X's are not X's those X's are Z's. That's the position along the propagation direction those are Z's since it's propagating in the Z direction alright now what the wave number is is it's related to the wave length it's just two Pi divided by the wave length and we know how to relate the angular frequency to things like the frequency and things like the period because we've done it many times before so let's do a quick example and get out here.

The following are the electric and magnetic fields let me leave equations, the following are the electric and magnetic fields that describe a particular electromagnetic wave what is the wavelength of this wave? What is the period? I'm going to address the period first just because what I am most comfortable with but it really doesn't matter how you do them because one answer does not depend upon the other this is the angular frequency in both cases. That's the quantity that we know is related to the wavelength sorry to the period we know that the angular frequency is 2 Pi over the period so we know that this is going to be 2 Pi over 4.19 times 10 to the 15 and so wops sorry I forgot a step there and so if I want to isolate the period I got to multiply the period up and divided the wavelength over so the period is 2 Pi over the wave length which is going to be 2 Pi over 4.19 times 10 to the 15 right that's the angular frequency and this whole thing is going to equal 1.5 times 10 to the -15 seconds which is an incredibly small number because the frequency is an incredibly large number we would expect the period to be very very very small if the frequency is very very very high now we want to address the wave number. We have our equation in red above in the green box for what the wave number is and we know that these numbers are both the wave numbers and the wave number is related to our wave length which is what we want to find so I can multiply the wavelength up and I can divide that over and I can get to the wavelength is 2 Pi over K. Which is 2 Pi over 1.4 times 10 to the 7. Which is going to be 450 or so times 10 to the -9 meters. I've chosen to write it like this for a particular reason because this can then be written as 450 nanometers which is blue light. So those are our two answers, now these two numbers have to actually be related to one another for multitude of reasons first of all we know that lambda divided by the period has to be the speed of the wave which is in this case light and you'll see that if you do take lambda and you do take the period and you divide them you will get C. That's one way to confirm that this is in fact a correct wave another way is to just take omega and divide it by K. Both of which are already represented in this equation, in the functions that will also equal the speed of light and finally we know that the ratio of E max to B max has to be the speed of light and if I wrote these functions correctly I check the multiple times I'm pretty sure I did all three of these relationships should be true. Alright guys that wraps up our discussion on electromagnetic waves as sinusoidal waves. Thanks for watching.

Electromagnetic Waves as Sinusoidal Waves Additional Practice Problems

The magnetic field of a plane EM wave is given by:  B = (E0/c) sin(kz + ωt) i  The electric field of the wave would be given by:

A. E = +E0 sin(kz + ωt) j

B. E = +E0 cos(kz + ωt) k

C. E = +E0 cos(kz + ωt) j

D. E = –E0 cos(kz + ωt) i

E. E = –E0 cos(kz + ωt) i

Watch Solution

The y component of the electric field of an electromagnetic wave travelling in the +x direction through vacuum obeys the equation Ey = (375 N/C) cos [kx − (2.20 × 10 14 rad/s)t]. What is the wavelength of this electromagnetic wave?

a. 0.272 μm

b. 1.36 μm

c. 2.72 μm

d. 8.57 μm

e. 17.1 μm

Watch Solution

A sinusoidal electromagnetic wave propogates in the -y direction in a medium with a refractive index of 1.5, with a magnetic field amplitude of 0.001 T and a wavelength of 1.55 nm. 

(a) What is the angular frequency of the wave?

(b) What is the electric field amplitude of the wave?

(c) What is the intensity of the wave?

Watch Solution

An electromagnetic wave propagates through a vacuum in the +x-direction, carrying an intensity of 150 W/m2. At t = 0, the electric field has zero amplitude, and after 0.01 s, the electric field strength grows to its maximum value, pointing in the +y direction. Write equations describing the electric and magnetic fields as sinusoidal oscillations, including the appropriate unit vectors to denote direction.

Watch Solution

At an instant of time and at a particular location in space, the electric field of an electromagnetic wave is in the -x-direction and the magnetic field is in the +y-direction. What is the direction in which the wave is traveling?

A) +x

B) -x

C) +y

D) -y

E) +z

F) -z

Watch Solution

At a certain instant in time, an electromagnetic wave has Ē in the +z direction and in the +y direction. In what direction does the wave propagate?

A) -x direction

B) +x direction

C) +y direction

D) -z direction

E) +z direction

Watch Solution