What Determines the Poynting Vector in an Electromagnetic Wave?

[waiting]

Here's the question:

An electromagnetic wave is traveling through vacuum. Its electric field vector is given by

E= E_0 sin(kx-wt)y_hat

where "y hat" is the unit vector in the y direction.

What is the Poynting vector S(x,t), that is, the power per unit area associated with the electromagnetic wave described in the problem introduction?

Please express in terms of all or some of the folowing variables: E_0, B_0, k,x,w,t, mu_0 and the unit vectors x_hat, y_hat and z_hat.

I've already tried (E_0 * B_0)/ mu_0...it didnt work, i think they want the answer in more depth!

Any help is appreciated!

 

[Hootenanny]

Here's the question:

An electromagnetic wave is traveling through vacuum. Its electric field vector is given by

E= E_0 sin(kx-wt)y_hat

where "y hat" is the unit vector in the y direction.

What is the Poynting vector S(x,t), that is, the power per unit area associated with the electromagnetic wave described in the problem introduction?

Please express in terms of all or some of the folowing variables: E_0, B_0, k,x,w,t, mu_0 and the unit vectors x_hat, y_hat and z_hat.

I've already tried (E_0 * B_0)/ mu_0...it didnt work, i think they want the answer in more depth!

Any help is appreciated!

Start by determining the corresponding magnetic field.

 

[baba89]

The Poynting vector, S(x,t), represents the flow of energy per unit area in the direction of propagation of an electromagnetic wave. It is given by the cross product of the electric field vector, E, and the magnetic field vector, B, divided by the permeability of free space, μ_0. In this case, the magnetic field vector is not given, but we can use the relationship between E and B in an electromagnetic wave to find it.

First, we need to find the magnetic field vector, B. According to Maxwell's equations, the relationship between E and B in vacuum is B = (1/c) * E, where c is the speed of light. Substituting the given electric field vector, E, we have B = (1/c) * E_0 sin(kx-wt)z_hat.

Now, we can calculate the Poynting vector using the formula S(x,t) = (1/μ_0) * E x B. Substituting the values for E and B, we have S(x,t) = (1/μ_0) * (E_0 sin(kx-wt)y_hat) x ((1/c) * E_0 sin(kx-wt)z_hat).

Using the properties of the cross product, we can expand this to S(x,t) = (1/μ_0c) * (E_0)^2 sin^2(kx-wt) (x_hat x z_hat). Since x_hat x z_hat = y_hat, we can simplify this to S(x,t) = (1/μ_0c) * (E_0)^2 sin^2(kx-wt) y_hat.

Therefore, the Poynting vector for this electromagnetic wave is S(x,t) = (1/μ_0c) * (E_0)^2 sin^2(kx-wt) y_hat, where E_0 is the amplitude of the electric field, k is the wavenumber, x is the position, w is the angular frequency, t is the time, μ_0 is the permeability of free space, and y_hat is the unit vector in the y direction.