Question

The magnetic field in a plane monochromatic electromagnetic wave with wavelength λ = 479 nm, propagating in a vacuum in the z-direction is described by B⃗ =(B1sin(kz−ωt))(i^+j^) where B1 = 10.4 X 10-6 T, and i-hat and j-hat are the unit vectors in the +x and +y directions, respectively.
1) What is k, the wavenumber of this wave?
2)What is zmax, the distance along the positive z-axis to the position where the magnitude of the magnetic field is a maximum at t = 0?
3)What is Emax, the amplitude of the electric field oscillations?
4)What is Ey, the y-component of the electric field at (x = 0, y-0, z = zmax) at t = 0?

Answer 1

Answer:

1)  k = 1.31 10⁻⁷ m, 2)     z = 59,875 (2n + 1) 10⁻⁹ m  3)   Ex = 3.12 10⁻³ N /C

Explanation:

The equation of a wave is

      y = A sin (kz -wt)

Where A is the amplitude, k the wave number and w the angular velocity.

1) we calculate the wave number

      k = 2π /λ

      k = 2π / 479 10⁻⁹

      k = 1.31 10⁻⁷ m

.2) the points where the magnetic field is maximum occurs when the sine function is maximum

       sin (kz - et) = + -1

       kz-wt = 90º = π/4  (2n + 1)         n integerr

       z = (π / 4) / k

       z = (π / 4) λ / 2π)

       z = λ/8  (2n + 1)

       z = 479 10⁻⁹ / 8 (2n + 1)

       z = 59,875 (2n + 1) 10⁻⁹ m

3) the electric and magnetic fields are related

        E = c B

Therefore we can calculate the maximum electric field from the magnetic field

        Bmax = B1 (i ^ + j ^)

         Ex = c B1 i ^

          Ex = 3 10⁸ 10.4 1⁻⁶

           Ex = 3.12 10⁻³ N /C

4)     E (x, y) = E1 sin (k z -wt)

       E1 = c B1

       E (x, y) = E1

Therefore the component Ey must be maximum

      Ey = c B1

      Ey = 3 108 10.4 10-6

      Ey = 312 10-3

Answer 2

The value of k is  $\rm 1.31\times10^{-7} \ m$,  $\rm z_{max}=1.199\times10^7$ meter, $\rm E_{max}= 22.06\times 10^2 \ N/c$, and  $\rm \left|\vec{_{E_y}} \right| =22.06\times10^2 \ N/c$

What is a magnetic field?

The magnetic field is defined as the field the magnetic materials generate or when an electric charge moves in a field region that generates the magnetic field.

We have a magnetic field in a plane monochromatic electromagnetic wave.

The equation for the wave is given by:

$\rm y = a \ sin (kz-wt)$

Where A = Amplitude of the wave

k = Wave-number

w = Angular velocity

For the k, we know:

$\rm k =\frac{2\pi}{\lambda}$

$\rm k =\frac{2\pi}{479\times10^{-9}}$      $\rm (\lambda = 479 \ nm = 479\times10^{-9} \ meter)$

$\rm k = 1.31\times10^{-7} \ m$

The point where the magnetic field is maximum:

$\rm sin(kz-wt)=\pm1$  or

kz - wt = π/2

kz  = π/2    ( t = 0)

$\rm z = \frac{\pi}{2k}$  

$\rm z = \frac{\pi}{2\times1.31\times10^{-7}}$    ($\rm k = 1.31\times10^{-7} \ m$)

$\rm z_{max}=1.199\times10^7$ m

For the $\rm E_{max}$  = $\rm c\times$$\rm B_{max}$    ( c is the speed of light)

$\rm B_1 = 10.4\times10^{-6 } \ T$

$\rm B_{max}= 10.4\times 10^{-6}\times\frac{1}{\sqrt{2} } \ T$

$\rm E_{max}=3\times10^8\times\rm 10.4\times 10^{-6}\times\frac{1}{\sqrt{2} }$

$\rm E_{max}= 22.06\times 10^2 \ N/c$

For  $\rm E_y$

$\rm \left|\vec{_{E_y}} \right| =\frac{cB_1}{\sqrt{2} }sin(kz_{max)}$

$\rm \left|\vec{_{E_y}} \right| =22.06\times10^2 \ N/c$

Thus, the value of k is  $\rm 1.31\times10^{-7} \ m$,  $\rm z_{max}=1.199\times10^7$ meter, $\rm E_{max}= 22.06\times 10^2 \ N/c$, and  $\rm \left|\vec{_{E_y}} \right| =22.06\times10^2 \ N/c$

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