Explaining Why Total Charge Induced = -q for Image Problem

Click For Summary

Discussion Overview

The discussion revolves around the image charge problem involving a conducting plane and a point charge. Participants explore the reasoning behind the total induced charge being equal to "-q," where q is the external point charge. The scope includes theoretical explanations and conceptual clarifications related to electrostatics and boundary value problems.

Discussion Character

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

Main Points Raised

  • One participant notes that the total charge induced on the conducting plane is "-q" and questions why this is equal to the image charge, suggesting that the field at distant points must cancel out for the total charge to sum to zero.
  • Another participant emphasizes that the tangential component of the electric field on the conducting surface must vanish, leading to the conclusion that an imaginary charge distribution of -q is necessary to maintain this condition.
  • A different perspective is offered, stating that image charges are not real but serve as a useful conceptual tool for solving boundary value problems in electrostatics.
  • One participant proposes using Gauss's law to assert that if the field corresponds to an image charge of -q, then this must also represent the total surface charge on the conductor.

Areas of Agreement / Disagreement

Participants express differing views on the reasoning behind the induced charge being -q, with some focusing on field cancellation and others on the conditions at the surface of the conductor. The discussion remains unresolved, with multiple competing explanations presented.

Contextual Notes

Participants do not fully agree on the implications of the image charge method, and there are varying interpretations of the physical principles involved, particularly regarding the role of the electric field and the nature of induced charges.

adityatandon
Messages
3
Reaction score
0
Consider the classic image problem with a conducting plane and a point charge. After finding out the charge density on the plane we integrate to find out the total charge induced. It comes out to be "-q",where q is the charge outside-
My book says "It comes out to be -q, as you can convince yourself with the benefit of hindsight". It isn't that obvious why it should come out equal to the image charge. Can somebody explain?
 
Physics news on Phys.org
adityatandon said:
Consider the classic image problem with a conducting plane and a point charge. After finding out the charge density on the plane we integrate to find out the total charge induced. It comes out to be "-q",where q is the charge outside-
My book says "It comes out to be -q, as you can convince yourself with the benefit of hindsight". It isn't that obvious why it should come out equal to the image charge. Can somebody explain?

At distant points the field due to the charge and its image cancel each other out. Now, since in fact the image is not there and the field is indeed caused by the induced charge, the sum of the induced charge and the point charge must be zero otherwise the field at distant points wouldn't be zero.
 
I would describe it a little bit different.
For me, the main point is that the tangential component of the electric field on the plate has to vanish. Otherwise, there would be movement of the electrons at the surface of the metal. So, the metal's surface is an equipotential surface of the electrostatic potential. Now you can think of an imaginary charge distribution that could be found such that the potential vanishes on the surface of the metal if the metal was not there. Of course, you find that it is a symmetrically placed charge -q. You can go even further and generalize this procedure and ask for metallic corners subject to a point charge etc.

The bottom line is that image charges are not real, just something that gives us some intuition about boundary value problems :)
 
Use Gauss's law. If the field is that from q'=-q, then that must also be the total surface charge.
 

Similar threads

High School Method of images and spherical coordinates
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad The Method of Images (Electromagnetism)
  • · Replies 4 ·
Replies
4
Views
4K
Graduate Work done via induced charges in a grounded conductor
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why Does Grounding Imply Zero Electric Potential in the Method of Images?
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Why is "method of image current" valid in magnetostatics?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Griffith, Electrodynamics, 4th Edition, Example 4.8. (First part)
  • · Replies 6 ·
Replies
6
Views
1K
Graduate [E&M] Question on the Image charge method of a grounding sphere
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Griffith, Electrodynamics, 4th Edition, Example 4.8. (Second part)
  • · Replies 3 ·
Replies
3
Views
967
Graduate The process of the method of images
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Image Method for B-Field Calculation on Iron Plate
  • · Replies 2 ·
Replies
2
Views
2K