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

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

Step-by-step 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=(2\pi)/(\lambda)`

where λ is the wavelength

So we get that

`k=(2\pi)/(\lambda) \rightarrow k=(2\pi)/(598 nm)  \rightarrow k=0,0105\ x\ 10^(9)\ m^(-1)`

The wavenumber is

`k=0,0105\ x\ 10^(9)\ m^(-1)` .