Electromagnetic plane waves

E_0*ω*θ*cos(ω/c*z-ωt) In summary, we have found the emf induced in a circular loop with radius a, N turns, and resistance R due to a plane electromagnetic wave traveling in vacuum. The emf is given by pi*a^2*E_0*ω*θ*cos(ω/c*z-ωt) and is dependent on the frequency, strength, and orientation of the electromagnetic wave, as well as the geometry of the loop. This solution assumes that a<<λ, meaning that the loop is much smaller than the wavelength of the electromagnetic wave.
  • #1
mattstover
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1. The problem statement
A plane electromagnetic wave traveling in a vacuum is given by E=(0, E_0*exp[i(kz-ωt)], 0) where E_0 is real. A circular loop of raduis a, N turns, and resistance R is located with its center at the origin. The loop is oriented so that a diameter lies along the z axis and the plane of the loop makes an angle θ with the y axis. Find the emf induced in the loop as a function of time. Assume that a<<λ.



Homework Equations





The Attempt at a Solution


attemp at a solution is attached as scan0001.jpg. the awnser should be

emf = pi*a^2*NkE_0*cos(theta)sin(wt)
 

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  • #2


Hello,

Thank you for posting this problem. Based on the information provided, it seems that the circular loop is experiencing a changing magnetic field due to the passing electromagnetic wave. This changing magnetic field will induce an electric field in the loop, causing an emf to be generated. To find the emf induced in the loop, we can use Faraday's Law, which states that the emf induced in a loop is equal to the negative rate of change of magnetic flux through the loop.

In this case, the magnetic flux through the loop will be changing due to the changing magnetic field of the electromagnetic wave. We can calculate the magnetic flux through the loop by taking the dot product of the magnetic field and the area vector of the loop. Since the loop is oriented so that its plane is perpendicular to the magnetic field, the dot product simplifies to B*A = B*A*cos(theta), where theta is the angle between the magnetic field and the normal to the loop's plane.

Substituting the given values, we have B = E_0*exp[i(kz-ωt)] and A = pi*a^2*sin(theta). Therefore, the magnetic flux through the loop is given by B*A = E_0*pi*a^2*sin(theta)*exp[i(kz-ωt)].

Now, we can take the negative time derivative of this magnetic flux to find the emf induced in the loop:

emf = -d(B*A)/dt = -E_0*pi*a^2*sin(theta)*iω*exp[i(kz-ωt)] = pi*a^2*E_0*ω*sin(theta)*exp[i(kz-ωt+π/2)]

Since we are only interested in the real part of this complex expression, we can drop the imaginary unit i and take the real part of the exponential term to get:

emf = pi*a^2*E_0*ω*sin(theta)*cos(kz-ωt)

This is the desired expression for the emf induced in the loop as a function of time. We can simplify it further by using the relation k = ω/c, where c is the speed of light in vacuum. This gives us:

emf = pi*a^2*E_0*ω*sin(theta)*cos(ω/c*z-ωt)

Since we are assuming that a<<λ, we can also use the small angle approximation sin(theta)
 

What are electromagnetic plane waves?

Electromagnetic plane waves are a type of electromagnetic radiation that has a constant electric and magnetic field. These waves travel in a straight line and do not require a medium to propagate, unlike mechanical waves like sound waves.

What is the speed of electromagnetic plane waves?

The speed of electromagnetic plane waves is the speed of light, which is approximately 3x10^8 meters per second. This speed is constant and does not change in a vacuum.

What are the properties of electromagnetic plane waves?

Electromagnetic plane waves have a wavelength, frequency, and amplitude. They also have an electric and magnetic field that oscillate perpendicular to each other and the direction of propagation.

How are electromagnetic plane waves produced?

Electromagnetic plane waves are produced when an oscillating electric charge or current is present. This can occur naturally, such as with lightning or the sun's radiation, or through man-made devices like radios and cell phones.

What are some applications of electromagnetic plane waves?

Electromagnetic plane waves have a wide range of applications in our daily lives, including communication (radio, television, and cell phones), medical imaging (MRI), and radar technology used in weather forecasting and air traffic control. They are also used in scientific research to study the properties of matter and the universe.

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