Magnetic field of the planar wave

[Andrei0408]

Homework Statement

The electric field for a planar electromagnetic wave propagating in a free space on Oz axis is 𝐸⃗ =
(𝐸0𝑥𝑖 + 𝐸0𝑦𝑗 ) 𝑠𝑖𝑛(𝜔𝑡 − 𝜅𝑧 + 𝜑). Find the magnetic field 𝐵⃗ of the planar wave.

Relevant Equations

Written on paper

I understand that because the vectors are perp, k x i = j, but why is k x j = -i? Why the minus? Could you please explain?

 

[Delta2]

There isn't much to explain, it follows from the definition of the unit vectors in cartesian system and from the definition of cross product that uses the right hand rule
Right-hand rule - Wikipedia

 

[Delta2]

I fail to see how you going to answer the question though.. Aren't you going to use the Maxwell-Faraday equation :
$$\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$$

 

[Andrei0408]

This is the way we solved it in class, is there another way?

 

[Delta2]

This is the way we solved it in class, is there another way?

Yes ok , apparently you use this formula $$\mathbf{B}=\frac{1}{c}\mathbf{u}\times\mathbf{E}$$ which is correct for plane EM waves with propagation vector ##\mathbf{u}##. It is a consequence from Maxwell-Faraday equation.

Actually it holds for all waves where the electric field is $$\mathbf{E}=\mathbf{E_0}f(\mathbf{u}\cdot\mathbf{r}\pm \omega t)$$ where ##\mathbf{E_0}## any constant vector and ##f## any real function of a real variable.

 

[Andrei0408]

Yes ok , apparently you use this formula $$\mathbf{B}=\frac{1}{c}\mathbf{u}\times\mathbf{E}$$ which is correct for plane EM waves with propagation vector ##\mathbf{u}##. It is a consequence from Maxwell-Faraday equation.

Thank you!