Question

Learning Goal: To practice Problem-Solving Strategy 32.1 Electromagnetic Waves. An electromagnetic wave in a vacuum traveling in the +x direction generated by a variable source initially has a wavelength λ of 305 μm and a maximum electric field Emax in the +y direction of 8.60×10−3 V/m . If the period of the wave is then increased by a factor of 1.70, what is the equation of the resulting magnetic field component of the wave?

Answer 1

Answer:

   B(z,t) = 2,867 10⁻¹¹ cos (2.06 10⁴ x - 3,635 10¹² t)

Explanation:

The wave equation for an electromagnetic wave is

            E = E₀ cos (kx –wt)

            B = B₀ cos (kx - wt)

Where these equations are for the electric and magnetic fields, respectively.

The two fields are related through the speed of light

               c = E₀ / B₀

Let's apply to our case, the wavelength is 305 μm = 305 10⁻⁶ m, The wave number is

           k = 2π / λ

           k = 2π / 305 10⁻⁶

           k = 2.06 10⁴ m

Period change to by a factor of 1.7, we write based on the initial period (T₀), the angular velocity is

          w = 2π f

With the initial data and the relationship

          c = λ f

         f = c / λ

        f = 3  10⁸/305 10⁻⁶6

         .f = 9.836 10¹¹ Hz

The frequency and period are related

         f = 1 / T

         T = 1.7 T₀

         f = 1 / 1.7 1 / T₀

         f = 1 /1.7  9.836 10¹¹

         f = 5,786 10¹¹ Hz

         w = 2π f

         w = 2π 5,786 10¹¹

        w = 3,635 10¹² rad / s

We look for the maximum amplitude of the magnetic field

           B₀ = E₀ / c

           B₀ = 8.60 10⁻³ / 3 10⁸

           B₀ = 2,867 10⁻¹¹ T

the fields and the velocity wave vector are perpendicular to each other, therefore the magnetic field oscillates in the z-direction

We build the equation of the magnetic field

          B(z,t) = 2,867 10⁻¹¹ cos (2.06 10⁴ x - 3,635 10¹² t)