# Finding the constants in an Electric Field Equation

## 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.

## 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:

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TSny
Homework Helper
Gold Member
Have you studied the properties of the "divergence" of the electric field?

vanhees71
Gold Member
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
Homework Helper
Gold Member
Have you studied the properties of the "divergence" of the electric field?
And how about the curl, since the field is apparently time-invariant.

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.

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

TSny
Homework Helper
Gold Member
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.