Question

A plane electromagnetic wave propagates in a vacuum in the z-direction. The wave has a wavelength of 10 m and the electric field is along the x direction and has an amplitude of is 0.20 V/m, with one maximum at x = 0 and t = 0. What is the mathematical expression for the magnetic field?

A. By = 0.067 × 10-8sin[(0.63 m-1)z - (2.1 × 10-9 s-1)t] T.
B. By = 0.067 × 10-8cos[(0.63 m-1)z + (1.9 × 108 s-1)t] T.
C. By = 0.067 × 10-8cos[(10 m-1)z - (2.1 × 10-9 s-1)t] T.
D. By = 0.067 × 10-8cos[(0.63 m-1)z - (1.9 × 10-8 s-1)t] T.
E. By = 0.067 × 10-8sin[(0.63 m-1)z + (2.1 × 10-9 s-1)t] T.

Answer 1

D.

`\mathbf{B_y = 0.067 * 10^(-8) Cos \Big[ 0.6 3m^(-1) z - (1.9 * 10^(-6) \ s^(-1))t \Big] T}`

Step-by-step explanation:

From the given information:

Let us recall that the direction with which electromagnetic wave proceeds is usually along
`E^( \to) * B^(\to)`

Thus; the magnetic field is always along the y-direction.

In the magnetic field, the maximum value of the field is expressed by using the formula:

`(E)/(C) = (0.20 \ V/m)/(3* 110^8 \ m/s) \\ \\ = 0.067 * 10^(-8) \ T`

Given that:

the maximum x is starting at 0, then, it implies that it is starting from the extreme position proceeding along +z direction.

As such; the general equation:
`y = A sin (\omega t - kz)`

Also:

`K = (2 \pi)/(\lambda)`

where;

`\lambda =10`

`K = (2 \pi)/(10)`

K ≅ 0.63 m⁻¹

Thus;

`\mathbf{B_y = 0.067 * 10^(-8) Cos \Big[ 0.6 3m^(-1) z - (1.9 * 10^(-6) \ s^(-1))t \Big] T}`

Answer 2

B. By = 0.067 × 10-8cos[(0.63 m-1)z + (1.9 × 108 s-1)t] T.

Step-by-step explanation:

Since the electric field has a maximum at x = 0 and t = 0, it is a cosine function and thus the magnetic field is also a cosine function.

Also, the electric field travels in the x - direction and the wave in the z-direction. Since the magnetic field is perpendicular to both directions, it must thus move in the y - direction.

So, By = B₀cos(kz - ωt). It has a negative sign since the wave is travelling in the positive z - direction.

Since c = E₀/B₀ where E₀ = amplitude of electric field = 0.20 V/m, B₀ = amplitude of magnetic field and c = speed of light = 3 × 10⁸ m/s

So, B₀ = E₀/c = 0.20 V/m ÷ 3 × 10⁸ m/s = 0.067 × 10⁻⁸ T

wave number, k = 2π/λ where λ = wavelength = 10 m. So, k = 2π/10 m = 6.28/10 m = 0.628 m⁻¹ ≅ 0.63 m⁻¹

angular frequency, ω = 2πf where f = frequency of wave = c/λ = 3 × 10⁸ m/s 10 m = 3 × 10⁷ s⁻¹. So, ω = 2πf = 2π(3 × 10⁷ s⁻¹) = 18.8 × 10⁷ s⁻¹ = 1.88 × 10⁸ s⁻¹ ≅ 1.9 × 10⁸ s⁻¹

Substituting the variables into By, we have

By = B₀cos(kz - ωt)

By = (0.067 × 10⁻⁸ T)cos[(0.63 m⁻¹)z - (1.9 × 10⁸ s⁻¹)t]

By = 0.067 × 10⁻⁸cos[(0.63 m⁻¹)z - (1.9 × 10⁸ s⁻¹)t] T

Since none of our options contain the given answer, we assume the wave moves in the negative z - direction. So, for that,

By = 0.067 × 10⁻⁸cos[(0.63 m⁻¹)z + (1.9 × 10⁸ s⁻¹)t] T