Finding a solution to Maxwell's equations from initial datas

[fluidistic]

Homework Statement

Suppose we know that B(\vec x ,t) is a solution to Maxwell's equations in vacuum and furthermore we know that E(\vec x , 0)=E_0.
How do we find E(\vec x , t)?

Homework Equations

\nabla \cdot E = 0.
\nabla \cdot B =0.
\vec \nabla \times \vec B = \left ( \frac{-1}{c} \right ) \cdot \frac{\partial E}{\partial t}
\vec \nabla \times \vec E = \left ( \frac{1}{c} \right ) \cdot \frac{\partial B}{\partial t}.
I'm using Gaussian's units.

The Attempt at a Solution

I think I could work with the 2 lasts equations I posted to find E but I don't reach anything. I'd like a very small guidance like if I'm in the right direction + a hint if possible.
Thanks.

 

[nickjer]

Think about integrating the 3rd equation with respect to time.

 

[fluidistic]

Think about integrating the 3rd equation with respect to time.

Thanks for the tip.
I reach \vec E=-c \int \vec \nabla \times \vec B dt. I think of using the initial condition. So E_0=-c\int \vec \nabla \times \vec B (\vec x ,0) dt.
I'm stuck here.

 

[nickjer]

The integral will have bounds.