Solving Image Problems with Orthonormal Systems - M. Next

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Discussion Overview

The discussion revolves around the concept of solving image problems using orthonormal systems in electrostatics, specifically involving point charges and their image charges in relation to various boundary conditions. Participants explore the implications of charge placement and symmetry in different configurations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a scenario with a point charge and its image charges, questioning the necessity of an additional charge when multiple images are created.
  • Another participant asserts that the net charge of the image charges should equal -q, suggesting that an additional +q charge is needed to balance the system.
  • A different participant challenges the assumption that the net charge must be -q and introduces a scenario where the axes are not perpendicular, raising questions about how to handle such cases.
  • One participant proposes a formula for determining the number of image charges based on the angle between the axes, indicating that only certain angles allow for this method to work.
  • Another participant seeks clarification on the equations and the placement of image charges, expressing confusion about the previous explanations.
  • A participant provides a visual representation of the problem, suggesting a method of reflection and symmetry to determine the placement of image charges.
  • Further clarification is requested regarding the reflection process and the sequence of adding charges, indicating a need for a clearer understanding of the method.
  • One participant expresses gratitude for the assistance and indicates that they have finally grasped the concept after further explanation.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of additional charges and the conditions under which the image charge method applies. The discussion remains unresolved regarding the implications of non-perpendicular axes and the specific placement of image charges.

Contextual Notes

Participants reference specific configurations and mathematical relationships without fully resolving the assumptions or limitations inherent in their approaches. The discussion includes various interpretations of the image charge method that depend on the geometry of the problem.

Who May Find This Useful

This discussion may be of interest to students and practitioners in electrostatics, particularly those exploring image charge problems and the mathematical frameworks used to analyze them.

M. next
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Hello :)
I would be very grateful if I get through with this bugging idea.
When I started reading about it, the first example was: A point charge in front of a finite plane conductor.
And I understood after some mathematical and geometrical procedure how it is okay to replace the finite plane with a charge of opposite sign.
Then I bumped into another problem:
Let us consider orthonormal system and point charge +q is placed on point (1,1) - it's image being in symmetry with respect to y-axis will be at (-1,1) and it's image abeing in symmetry with respect to x-axis will be at (1,-1). Until here I was okay, till I realized that another point charge was placed at (-1,-1). Why?
What's the whole point?

Thanks in advance.

M. next
 
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The net charge of the image charges should be -q shouldn't it? If you add the two charges that you made at (-1,1) and (1,-1), those would add up to be -2q. Thus, you need 1 more +q charge at (-1,-1).
 
Who said that the net charge must be -q?
And another thing: if y-axis and x-axis are not perpendicular but they have an acute angle between them (say 60) and q is placed at (4,1) as in the old orthonormal system (am only placing coordinates to clear up the image). What will happen? What should we do then? How do we treat this case?
 
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If your angle is theta then the number of image charges you would need would be 360/theta -1. This number must be an integer so only angles that divide 360 are allowed for this method to work.
 
nucl34rgg, I appreciate it a lot. But can you please tell me how do you know these equations? Because I read the whole course and didn't come across these equations.
And then you told me the nb of images but you didn't tell me where to locate them?
 
Ok this picture is embarrassingly bad, but it will serve the purpose. Pretend each angle between the lines is 60 degrees, and pretend that the figure is radially symmetric. All you have to do is reflect around and go around in a circle. Treat the one +q in the top right as real and the rest are image charges. Then the lines along the boundary for that +q have V=0.
 

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Thanks for your time, I really appreciate it. But what do you mean by "reflect around and go around in a circle"
You must be fed up by now, but I have to understand this by tomorrow.
Thanks again
 
start with the positive q charge on the top right. reflect over the v=0 boundary line. add a -q charge there and reflect over the next line...add a +q charge there, etc
 

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Oh now i get it! Finally!
Thank you loads nucl34rgg. I really appreciate your time, and patience. :)
 
  • #10
Glad to help! :)
 

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