Calculate Magnetic field

[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 $$\vec{B}=\left(\begin{array}{c} \frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3} \end{array}\right) \times \left(\begin{array}{c} A_{01} ~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \\ A_{02} ~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \\ A_{03} ~e^{-i(k_1x_1+k_2x_2+k_3x_3-\omega t)} \end{array}\right).$$