The magnetic field of an electromagnetic wave traveling along the z-direction is described by the wave function B ( z , t ) = 1.0 × 1 0 − 3 sin ( k z − 1.27 × 1 0 12 t ) B\left(z,t\right)=1.0\times10^{-3}\sin\left(kz-1.27\times10^{12}t\right) B ( z , t ) = 1.0 × 1 0 − 3 sin ( k z − 1.27 × 1 0 12 t ) , where k is the wave number. Write the complete wave function for the electric field of this wave.

Alright, folks. So, hopefully, you got a chance to work this one out on your own. So the magnetic field of this traveling electromagnetic wave is described by this wave function over here. So we've got b, which is, oscillating in the z direction, so we've got the z here, and we've got 1 times 10 to the minus 3 times sine of kz minus 1 0.27 times 10 to the 12 t, where it tells us that k is just some wave number. Right? So we don't know what actually that wave number is, but it is a number. And what we're supposed to do is we're supposed to find the complete wave function for the electric field of this wave. So this is actually very similar to a problem that we've already done before, but sort of just backwards. Instead of writing the magnetic field, we're gonna be writing the electric field. Alright? So let's get started. Remember that the form of an electric field is something like this. So we have that e in the z direction, so ez of t, is equal to this is going to be times the sine of kz minus omega t. So we need three things to write the complete wave function for this wave. We have to have k max, we have to have k or sorry, k e max, and then k, and then we also have to have omega. Now remember that the, the things in the parentheses are always going to be the same for these two wave functions. So this little piece here, 1.27 times 10 to 12 is actually what our omega is. Alright? So that's the same for both. So we already have what omega is. Now we just need to find what e max and k are so we can write our complete wave function. Let's start off with the first one here, which is E max. How do we figure that out? Remember, whenever we want to relate the strengths of the magnetic field and electric field, we always used to use the equation that e max, is equal to c times b max. So we use c times b max, which we just get from our table over here, this equation. Alright? So we have that e max, if we just plug in some numbers here, is just going to be 3 times 10 to the 8th, times the b max, and the b max in this case is just the the amplitude of this wave function. This is here our b max. Alright? So this is just gonna be our 1 times 10 to the minus 3. Now when you work this out, what you should get here is that e max is equal to 3 times 10 to the 5th, and that's gonna be in Newtons per Coulomb. Alright? So that's your e max. So that's the first thing. Alright. So now let's move on to the second part here, which now we wanna figure out what our k is. So let's just do that over here. So this is our e max, and let's figure out what our k is. Alright. So for k, remember that k has a couple of different, couple of different equations. We have k is equal to 2 pi over lambda. So can we use this? Well, the problem with this one is we actually don't know what lambda is. We don't know what the wavelength is. However, we do know what omega is. And because of that, we're actually just gonna use this bottom equation over here, which relates omega to k by, basically, just the 2, left and right equations and ignoring the one that's in the middle here. Alright? So we're gonna ignore this one, and we're just gonna focus on this expression. So we have that omega is equal to c times k. And so therefore, all we have to do is just divide by c, and this gives us k, and we just use 1.27 times 10 to the 12th divided by the speed of light which is 3 times 10 to the 8th. When you work this out what you should get is you should get a k value of 4233. Remember k doesn't have any units. Alright? So now, we actually have everything that we need to write out our wave function. We've got Emax, we solve for that already, and we also have k, we solve for that. So that means we just bring this down here. Our final expression should be e z t is equal to our emax, which is 3 times to the 5th. This is gonna be sine of, and then we're gonna use k, which is 4233 x minus or actually that's not x, it's z minus, and then this is gonna be omega, which is 1.23, 27 times 10 to 12th, and this is gonna be t. Alright? So this is actually our complete wave function for the electric field of this electromagnetic wave. Let me know if you have any questions. Hopefully, you got it right, and I'll see you in the next video.

32. Electromagnetic Waves

Wavefunctions of EM Waves

Physics

32. Electromagnetic Waves

Wavefunctions of EM Waves

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Multiple Choice

The magnetic field of an electromagnetic wave traveling along the z-direction is described by the wave function $B\left(z,t\right)=1.0\times10^{-3}\sin\left(kz-1.27\times10^{12}t\right)$ , where k is the wave number. Write the complete wave function for the electric field of this wave.

$E\left(z,t\right)=1.0\times10^3\sin\left(3.81\times10^{20}x-1.27\times10^{12}t\right)$

$E\left(z,t\right)=3.0\times10^{11}\sin\left(3.81\times10^{20}x-1.27\times10^{12}t\right)$

$E\left(z,t\right)=3.0\times10^5\sin\left(4.23\times10^3x-1.27\times10^{12}t\right)$

$E\left(z,t\right)=1.0\times10^3\sin\left(4.23\times10^3x-1.27\times10^{12}t\right)$

Verified step by step guidance

Identify the given wave function for the magnetic field: B(z,t) = 1.0 \times 10^{-3} \sin(kz - 1.27 \times 10^{12}t).

Use the relationship between the electric field and magnetic field in electromagnetic waves: E_{max} = cB_{max}, where c is the speed of light (3.0 \times 10^8 \text{ m/s}).

Calculate the maximum electric field (E_{max}) using the given B_{max} = 1.0 \times 10^{-3} and c = 3.0 \times 10^8 \text{ m/s}.

The wave number k is related to the wavelength \lambda by k = \frac{2\pi}{\lambda}. Since the angular frequency \omega = 1.27 \times 10^{12} \text{ rad/s} is given, use the relationship \omega = ck to find k.

Substitute the calculated E_{max} and k into the electric field wave function: E(z,t) = E_{max} \sin(kz - 1.27 \times 10^{12}t) to write the complete wave function for the electric field.

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