Question

A circular loop of wire has radius 7.80cm . A sinusoidal electromagnetic plane wave traveling in air passes through the loop, with the direction of the magnetic field of the wave perpendicular to the plane of the loop. The intensity of the wave at the location of the loop is 2.03�10?2W/m2 , and the wavelength of the wave is 6.20m .What is the maximum emf induced in the loop? Express your answer with the appropriate units.I stumbled through the formulas I do know for EMF but cant seem to figure out how to get the right answer. Please help and provide explanation! Thanks

Answer 1

Answer:

fem = -  4.50   10²² V

Explanation:

For the solution of this problem we must use the equation of the induced electromotive force or Faraday's law

        E = - d Φ$._{B}$ / dt = d (BA cos θ) dt

In this case they tell us that the magnetic field is perpendicular to the plane of the loop, as the normal to the surface of the loop is in the direction of the radius, the angle  between the field and this normal is zero, so cos 0º = 1. The area of ​​the loop is constant, with this the equation is

        E = - A dB / dt       (1)

To find field B, we have the relationships of electromagnetic waves

       E = c B

The intensity or poynting vector for the wave is described by the equation

       S = I = 1 / μ₀ E x B = 1 /μ₀ E B

We replace

      I = 1 /μ₀  (cB) B = c /μ₀ B²

This is the instantaneous intensity.

      B = √ (μ₀ I /c)

We substitute in equation 1

      E = - A μ₀/c d I / dt

With the maximum value we are asked to change it derived from variations

      E = -A c/μ₀ ΔI / Δt

It remains to find the time of the variation. Let's use the equation

      c = λ f = λ / T

      T = λ / c

      T = 6.20 / 3 10⁸

      T = 2.06 10⁻⁸ s

We already have all the values ​​to calculate the fem

      fem = - π r² c/μ₀ ΔI/Δt

     fem = - (π 0.078²)  (3 10⁸/(4π 10⁻⁷)  (2.03 10² -0) / (2.06 10⁻⁸ - 0)

     fem = -  4.50   10²² V