Question

The magnetic field in a plane monochromatic electromagnetic wave with wavelength λ = 598 nm, propagating in a vacuum in the z-direction is described by B =(B1sin(kz−ωt))(i^+j^) where B1 = 8.7 X 10-6 T, and i-hat and j-hat are the unit vectors in the +x and +y directions, respectively. What is k, the wavenumber of this wave?

Answer 1

Answer:

For this given plane monochromatic electromagnetic wave with wavelength λ=598 nm, the wavenumber is $k=0,0105\ x\ 10^{-9}\ m^{-1}$ .

Explanation:

For a plane electromagnetic wave we have that the electrical and magnetic field are:

$E(r,t)=E_{0}\ cos ( wt-kr)\\\ B(r,t)=B_{0}\ cos(wt-kr)$

In this case we have the data for the magnetic field. We are told that the magnetic field in a plane electromagnetic wave with wavelength λ=598 nm, propagating in a vacuum in the z direction ($\hat k$) is described by

         $B=8.7\ x\ 10^{-6}\ T sin(kz-wt) (\hat i+\hat j)$

($\hat i,\hat j, \hat k$ are the unit vectors in the x,y,z directions respectively)

The wavenumber k is a measure of the spatial frequency of the wave, is defined as the number of radians per unit distance:

          $k=\frac{2\pi}{\lambda}$

where λ is the wavelength

So we get that

$k=\frac{2\pi}{\lambda} \rightarrow k=\frac{2\pi}{598 nm} \rightarrow k=0,0105\ x\ 10^{9}\ m^{-1}$

The wavenumber is

            $k=0,0105\ x\ 10^{9}\ m^{-1}$ .