Suppose that a right-moving EM wave overlaps with a left-movin...

Question

Suppose that a right-moving EM wave overlaps with a left-moving EM wave so that, in a certain region of space, the total electric field in the y direction and magnetic field in the z direction are given by Eᵧ = E₀ sin(kx - ωt) + E₀ sin(kx + ωt) and Bz = B₀ sin(kx - ωt) - B₀ sin(kx + ωt). Determine the Poynting vector and find the x locations at which it is zero at all times.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with the pointing vector. This problem says, suppose that a standing wave is formed by two identical electromagnetic waves, one propagating upwards along the Z axis and the other propagating downwards along the same axis. The interference pattern results in the following standing wave expression. For the electric and magnetic fields we have ex is equal to two E knot multiplied by the sine of KZ multiplied by the cosine of omega T and by is equal to minus two B not multiplied by cosine of KZ multiplied by the sine of omega T. The pointing vector which represents the energy flow of the electromagnetic wave is defined as S is equal to one divided by mu knot multiplying E cross B where E is in the X direction and B is in the Y direction. And so the S will be in the Z direction. We need to determine the pointing vector S and identify the Z positions where the pointing vector is zero at all times. We're given four possible choices as our answers. For choice A we have S is equal to minus E knot multiplied by B not divided by Mu knot. And that fraction is multiplied by the sine of quantity of two KZ in quantity. And that's multiplied by the sine of the quantity of two omega T in quantity. K hat nz is equal to N pi divided by two K where N is equal to zero plus or minus one plus or minus two and so forth. And that's gonna be the same end value for all the answers. Now, for choice B we have S is equal to minus E not multiplied by B not divided by Muno. And that's multiplied by the sine of two Kz. And that's multiplied by the sine of two Omega TK hat and Z is equal to N multiplied by pi divided by or K. For choice C, we have S is equal to E knot multiplied by B not divided by Muno. And that's multiplied by the sine of two KZ multiplied by the sine of two Omega TK hat and Z is equal to N pi divided by two K. And finally, for choice D, we have S is equal to E knot multiplied by B knot divided by mu knot. And that fraction is multiplied by the sine of two KZ. And that's multiplied by the sine of two Omega TK hat NZ is equal to N pi divided by four K. So the question wants us to calculate the pointing vector. So we'll use the definition that we're given in the problem and that is S is equal to one divided by Muno multiplied by E cross B. We were given expressions for the electric field and the magnetic field. And we're told that the electric field is only in the X direction and the magnetic field is only in the Y direction. So we'll plug those um expressions in for the electric and magnetic fields. And so we'll have s is equal to one divided by Muno, that's multiplying the quantity of two eno multiplied by the sign of KZ multiplied by the cosine of omega T in quantity. And that is actually in the X direction. And so that's gonna be going in the IHAT direction and this is gonna be crossed with the quantity or RB that is going to be minus two B not multiplied by cosine of KZ multiplied by the sign of Omega T in quantity. And that's gonna be in the J hat direction since it's along with the Y direction. So here, now I can just do the cross product. And so I'll multiply the coefficients together. And so I'll get S is equal to four or minus four multiplied by en multiplied by BN divided by M not, that's gonna be multiplied by the sine of KZ multiplied by the cosine of Daisy. And then that's gonna be multiplied by the sign of Omega T multiplied by the cosine of Omega T. And then I'll have I hat cross J hat and that will give me K hat I can simplify this a little bit by using our uh double angle formula or sign. And so recall that the sign of two X is equal to two multiplied by the sine of X multiplied by the cosine of X. So we can use that to combine the sine and cosine for the KZ argument and do the same for the omega T argument for those sine and cosine terms. And so to do that, we'll um actually pull a factor of two from the four out in front to go with the sine and cosine for the KZ term. And so we'll have S is equal to minus E not multiplied by B knot divided by mu knot. And then we'll have the sign of two KZ. And that's gonna be multiplied by the quantity of sign of two omega T multiplied by K hat. And so for the mega um T terms cosine and sine terms there, we combine them also pull another factor of two from the four out in front. And so that's gonna be our answer for the pointing vector. Now, for the second half of this question, we need to figure out where this pointing vector is going to be zero at all times. And that's going to correspond to when the sign of two KZ is equal to zero. That means that the argument inside there the two KZ has to be an integer, multiple of pi, so we'll have two KZ is equal to N multiplied by pi where N here can be equal to zero plus or minus one or plus or minus two and so forth. So we just need to solve this expression for Z by dividing both sides by two K. And then we'll get Z as equal to N pi divided by the quantity of two K. So that's gonna be the second half of our answer. So now when we look back at our answer choices, the correct answer choice corresponds to choice A. So to solve this problem, we just had to directly apply our formula for the pointing vector substituting in the electric and magnetic fields that we were given in the problem. In order to simplify it, we did have to use some trick a trig identity in order to combine our sine and cosine terms. And we actually had to combine this sine and cosine terms in order to figure out the locations where the pointing vector is going to be zero at all times. And for that, we just had to set our argument of our sine function, which in this case is two KZ equal to an integer, multiple pi which would cause that sine function to be zero regardless of what the time is. So I hope that this has been useful. And I'll see you in the next video.

Key Concepts

Electromagnetic Waves

Superposition Principle

Poynting Vector

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