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?

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)