Question

Write equations for both the electric and magnetic fields for an electromagnetic wave in the red part of the visible spectrum that has a wavelength of 700 nm and a peak electric field magnitude of 3.5 V/m. (Use the following as necessary: t and x. Assume that E is in volts per meter, B is in teslas, t is in seconds, and x is in meters. Do not include units in your answer. Assume that E = 0 and B = 0 when x = 0 and t = 0.) E(x, t) = B(x, t) =

Answer 1

`E=3.5(8.98*10^(6)x-2.69*10^(15)t)`

`B=1.17*10^(-8)(8.98*10^(6)x-2.69*10^(15)t)`

Step-by-step explanation:

The electric field equation of a electromagnetic wave is given by:

`E=E_(max)(kx-\omega t)` (1)

• E(max) is the maximun value of E, it means the amplitude of the wave.
• k is the wave number
• ω is the angular frequency

We know that the wave length is λ = 700 nm and the peak electric field magnitude of 3.5 V/m, this value is correspond a E(max).

By definition:

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

`k=8.98*10^(6) [rad/m]`

And the relation between λ and f is:

`c=\lambda f`

`f=(c)/(\lambda)`

`f=(3*10^(8))/(700*10^(-9))`

`f=4.28*10^(14)`

The angular frequency equation is:

`\omega=2\pi f`

`\omega=2\pi*4.28*10^(14)`

`\omega=2.69*10^(15) [rad/s]`

Therefore, the E equation, suing (1), will be:

`E=3.5(8.98*10^(6)x-2.69*10^(15)t)` (2)

For the magnetic field we have the next equation:

`B=B_(max)(kx-\omega t)` (3)

It is the same as E. Here we just need to find B(max).

We can use this equation:

`E_(max)=cB_(max)`

`B_(max)=(E_(max))/(c)=(3.5)/(3*10^(8))`

`B_(max)=1.17*10^(-8)T`

Putting this in (3), finally we will have:

`B=1.17*10^(-8)(8.98*10^(6)x-2.69*10^(15)t)` (4)

I hope it helps you!