Finding the constants in an Electric Field Equation

  • #1

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)[itex]\widehat{a}_{x}[/itex]+B(x-y)[itex]\widehat{a}_{y}[/itex]]cos(wt)

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

20130927_151144_zps7427c05e.jpg


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:

20130927_154159_zps9ef8919b.jpg


Can someone please help point me in the right direction?
 

Answers and Replies

  • #2
TSny
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Have you studied the properties of the "divergence" of the electric field?
 
  • #3
vanhees71
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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 [itex]A[/itex] and [itex]B[/itex] as much as you can.
 
  • #4
rude man
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Have you studied the properties of the "divergence" of the electric field?
And how about the curl, since the field is apparently time-invariant.
 
  • #5
Using Maxwell's equation

[itex]\nabla \cdot E[/itex]=[itex]\frac{q_{ev}}{ε_{0}}[/itex]

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

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

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

(A-B)cos(ωt)=0

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

I am not really sure how to interpret this result.
 
Last edited:
  • #6
Sorry I had some mistakes in there, I was making a few changes!
 
  • #7
TSny
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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.
 
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