Calculate E and B fields when given A

[Lambda96]

Homework Statement

Calculate $\vec{B}(t,\vec{x})$

Relevant Equations

none

Hi

I'm not sure if I calculated the magnetic field from task a) correct?

for calculatin $\vec{B}$ i used, the formular $\vec{B}=\vec{\nabla} x \vec{A}$

$$\vec{B}=\left(\begin{array}{c} \frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3} \end{array}\right) \times \left(\begin{array}{c} A_0\cdot e^{-i(k_1x_1-\omega t)} \\ A_0\cdot e^{-i(k_2x_2-\omega t)} \\ A_0\cdot e^{-i(k_3x_3-\omega t)} \end{array}\right)=\left(\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right)$$

Is that right?

 

[kuruman]

It looks like you misinterpreted the components of the vector potential.

 

[Lambda96]

Thank you kuruman for your help

Should the calculation look like this?

$$\vec{B}=\left(\begin{array}{c} \frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3} \end{array}\right) \times \left(\begin{array}{c} A_x\cdot e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \\ A_y\cdot e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \\ A_z\cdot e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \end{array}\right)$$

 

[kuruman]

Much better, but it seems that you think that $~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)}~$ is some kind of a vector. It is not and no \cdot is needed on the right hand side.

Also, since you are given the constant vector $\mathbf A_0$ and you are using numbers as subscripts, it would be consistent to write it as $$\vec{B}=\left(\begin{array}{c} \frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3} \end{array}\right) \times \left(\begin{array}{c} A_{01} ~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \\ A_{02} ~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \\ A_{03} ~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \end{array}\right).$$

 

[Lambda96]

Thank you kuruman for your help , I have now adjusted the notation in my submission

 

[Orodruin]

To add to what has already been said, you seem to insist in writing out all of your expressions on component form. While this may be instructive in the beginning, we invented vector notation precisely to not have to do this. Your entire computation would be simpler in vector notation using the vector identity $\nabla \times (\phi \vec V) = \phi \nabla \times \vec V - \vec V \times \nabla \phi$, where $\vec V$ is an arbitrary vector field and $\phi$ is an arbitrary scalar field.