Finding the magnetic field B given the vector potential A ?

[patric44]

Homework Statement

find B given the vector potential A

Relevant Equations

A = /hat(ρ)/ρ *φ(z,t)

hi guys
this seems like a simple problem but i am stuck reaching the final form as requested , the question is
given the magnetic vector potential
$$\vec{A} = \frac{\hat{\rho}}{\rho}\beta e^{[-kz+\frac{i\omega}{c}(nz-ct)]}$$
prove that
$$B = (n/c + ik/\omega)(\hat{z}×\vec{E})$$
simple enough i found the curl of A which came out to be :
$$B = \frac{\hat{\phi}}{\rho} \beta e^{-iwt} (-k+\frac{i\omega n}{c})e^{-kz+\frac{i\omega n z}{c}}$$
how do i reach the given form of B ?!

 

[TSny]

How is $\vec E$ related to $\vec A$? Are you given any information about the scalar potential $\varphi$?

 

[patric44]

How is $\vec E$ related to $\vec A$? Are you given any information about the scalar potential $\varphi$?

no nothing else was given ?! , could it be related some how to maxwell third equation $∇×E = -\frac{∂B}{∂t}$

 

[TSny]

no nothing else was given ?! , could it be related some how to maxwell third equation $∇×E = -\frac{∂B}{∂t}$

See here for how $\vec E$ is related to $\vec A$ and $\varphi$. There is some freedom in the choice of $\vec A$ and $\varphi$ ("gauge freedom"). Maybe this problem is assuming a choice of gauge where $\varphi = 0$ at each point of the region of interest, which can be done if there is no charge in the region.