Finding the constants in an Electric Field Equation

[CanIExplore]

Homework Statement

The problem calls for finding the relationship between the constants A and B in the following equation of an instantaneous electric field:

E=[A(x+y)\widehat{a}_{x}+B(x-y)\widehat{a}_{y}]cos(wt)

The details of the medium in which the field exists are that it is homogenous, isotropic, linear, and is source-free.

Homework Equations

The Attempt at a Solution

I really have no idea how to go about solving this problem. Initially I was thinking I was just playing around with the vector equation by seeing what it looks like when I plug in different values of x and y and when t=0 or 1. Here is what that looked like:

Can someone please help point me in the right direction?

 

[TSny]

Have you studied the properties of the "divergence" of the electric field?

 

[vanhees71]

The only question is, what an "instantaneous" electric field might be. I think it's just an electric field, right?

Then just use the Maxwell and constitutive equations for the given situation to constrain the functions A and B as much as you can.

 

[rude man]

Have you studied the properties of the "divergence" of the electric field?

And how about the curl, since the field is apparently time-invariant.

 

[CanIExplore]

Using Maxwell's equation

\nabla \cdot E=\frac{q_{ev}}{ε_{0}}

where q_{ev} is the electric charge density and noting that q_{ev}=0 for a source free medium, I get:

\nabla \cdot E={\frac{\partial}{\partial_{x}}\widehat{a}_{x}\cdot[\widehat{a}_{x}A(x+y)]+\frac{\partial}{\partial_{y}}\widehat{a}_{y}\cdot[\widehat{a}_{y}B(x-y)]}cos(ωt)=0

={\frac{\partial}{\partial_{x}}Ax+\frac{\partial}{\partial_{x}}Ay+\frac{\partial}{\partial_{y}}Bx-\frac{\partial}{\partial_{y}}By}cos(ωt)=0

(A-B)cos(ωt)=0

Which is true when ωt=\frac{\pi}{2}, or when A=B.

I am not really sure how to interpret this result.

 

[CanIExplore]

Sorry I had some mistakes in there, I was making a few changes!

 

[TSny]

You might want to start with what you know about $\vec{\nabla} \cdot \vec{D}$ and then, as vanhees71 suggested, use the consitutive relations to relate $\vec{D}$ to $\vec{E}$.

Using all four of the assumptions: (1) source free (2) homogeneous (3) isotropic (4) linear medium, you can deduce what $\vec{\nabla} \cdot \vec{E}$ must equal.