How Does Faraday's Law Apply to EM Plane Waves in Conducting Loops?

[jegues]

Homework Statement

See figure attached.

The file attach isn't working so here is a link to the problem statement,

http://imgur.com/FhoIe

The answer is given in a red box.

Homework Equations

The Attempt at a Solution

The file attach isn't working so here is a link to my attempt at the solution,

http://imgur.com/0nkaC

I'm fairly certain I'm not making correct use of the information given about the radiated power in order to solve this problem.

It is also possible that I've made mistakes elsewhere as well.

What am I doing wrong?

 

[BruceW]

neither of the links are working for me...

 

[jegues]

neither of the links are working for me...

Let me try attaching them to this post.

 

[BruceW]

Those are working. But the one with your writing on is a little difficult to read.

Anyway, about the problem, you start off with the equation for the magnetic flux through the area of the coil, which is good, this does cause the emf. But then you start using the notation H_yⁱ I'm not sure what this means. From the context, it looks like it should be the magnetic field, right?

And then you give the radiated power as the real part of the square of the electric field, divided by something that looks like the symbol nu. What is the symbol meant to represent?

 

[asteriatic]

I would first start by reviewing the problem statement and making sure I understand all of the given information and equations involved. From the figure provided, it appears that we are dealing with an electromagnetic plane wave traveling in the x-direction with an electric field given by E=E0cos(kx-wt). This wave is incident on a conducting loop, which is connected to a resistor and a battery, creating a closed circuit. Faraday's law states that the induced EMF in a closed loop is equal to the negative rate of change of magnetic flux through the loop, or EMF=-dΦ/dt. In this case, the magnetic flux through the loop is changing due to the varying electric field of the incident wave, which creates a changing magnetic field according to Maxwell's equations.

To solve this problem, we need to find the induced EMF in the loop, which can be calculated using Faraday's law. From the given information, we know that the loop has a length of L and a width of a, and that the incident wave has a frequency of ω and a wave number of k. We can then calculate the magnetic flux through the loop by integrating the magnetic field over the surface of the loop, which is given by B=B0cos(kx-wt). This yields a magnetic flux of Φ=B0aLcos(ωt). Taking the derivative of this with respect to time, we get dΦ/dt=-B0aLωsin(ωt).

Now, we can plug this into Faraday's law to find the induced EMF. We know that the EMF is equal to the negative of the rate of change of magnetic flux, so EMF=-dΦ/dt=B0aLωsin(ωt). This EMF will induce a current in the loop, which will create a magnetic field that opposes the change in the incident magnetic field. This is known as Lenz's law, and it ensures that energy is conserved in the system.

To find the power radiated by the incident wave, we can use the Poynting vector, which gives the power per unit area carried by an electromagnetic wave. The Poynting vector is given by S=1/μ0(E x B), where μ0 is the permeability of free space. In this case, the electric and magnetic fields are perpendicular to each other, so the vector product simplifies to S