Construct B field from a given E field using Maxwell's Eqns

[Robsta]

Homework Statement

Given an electric field in a vacuum:
E(t,r) = (E₀/c) (0 , 0 , y/t²)

use Maxwell's equations to determine B(t,r) which satisfies the boundary condition B -> 0 as t -> ∞

Homework Equations

The problem is in a vacuum so in the conventional notation J = 0 and ρ = 0 (current density is zero and charge density is zero). Maxwell's equations are reduced to:

Div(E) = 0
Div(B) = 0
Curl(E) = -∂B/∂T
Curl(B) = μ₀ε₀∂E/∂t

The Attempt at a Solution

I've been given E so taking the curl of it I get

Curl E = (E₀/c) (t^-2 , 0 , 0)

So using the third Maxwell Equation I get ∂B/∂T = -(E₀/c) (t^-2 , 0 , 0)

But this is a partial differential. When I integrate the components of the vector with respct to t, I have to add functions of x,y,z.

So B = E₀/c (1/t + f(x,y,z) , g(x,y,z) , h(x,y,z) )

Taking the curl of this vector is tricky but I'll give it a go.

Setting the curl of my B vector above to equal ε₀μ₀E I get a big mess of partial derivatives of f(x,y,x), g(x,y,z) and h(x,y,z) equaling my original E vector, which two of its components are zero and one is yt_-2

Help please?

 

[BvU]

Why would you want to take the curl of B ?

And you have a boundary condition for B that gets rid of all these f, g and h

 

[Robsta]

Oh of course. Since B tends to zero with long times, there can't be any steady B fields like f, g and h would necessitate. Thanks very much!