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.

 

[paranormal]

I would suggest approaching this problem by first reviewing the fundamental principles of Maxwell's equations and their solutions. It is important to understand the physical meaning of each equation and how they relate to each other.

From the given information, we know that B(\vec x ,t) is a solution to Maxwell's equations in vacuum, which means it satisfies all four equations simultaneously. We also know that E(\vec x , 0)=E_0, which gives us the initial condition for the electric field at time t=0.

To find E(\vec x , t), we can start by considering the equation \vec \nabla \times \vec E = \left ( \frac{1}{c} \right ) \cdot \frac{\partial B}{\partial t}. This equation relates the time derivative of the magnetic field to the curl of the electric field. Since we know the solution for B(\vec x ,t), we can use this equation to find the time evolution of E(\vec x , t).

Next, we can use the equation \nabla \cdot E = 0 to determine the divergence of the electric field. This equation tells us that the electric field must have zero divergence in vacuum.

Using these two equations, we can work towards finding the solution for E(\vec x , t). It may also be helpful to consider the physical meaning of the electric field and how it relates to the magnetic field in terms of electromagnetic waves.

Overall, my advice would be to carefully review the equations and their physical interpretations, and then systematically apply them to find the solution for E(\vec x , t). It may also be helpful to consult with other scientists or resources for additional guidance and support.