B Symmetry regarding induced potentials?

AI Thread Summary
The discussion centers on the relationship between induced potentials in electrostatics, specifically regarding a point charge and an uncharged conductor. The referenced paper suggests that the potentials due to these charges, denoted as ##\bar\phi_x(y)## and ##\bar\phi_y(x)##, are equal, prompting questions about the validity of this assertion. Participants clarify that this relationship can be derived from Green's Reciprocity Theorem, which applies under the assumption of vanishing charge densities at infinity. Additionally, it is noted that while the induced surface charge on the conductor must be considered, its contribution to the potential integration is zero due to the conductor being an equipotential surface.
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2024 Award
Messages
42,656
Reaction score
10,435
TL;DR Summary
Is it true that the potential arising at A from induced charges on conductor B from point charge q at C = that arising at C from induction on B from charge q at A?
A homework thread, https://www.physicsforums.com/threa...etal-sheet-along-a-spherical-surface.1057702/, references https://arxiv.org/pdf/1007.2175.pdf.
There is an uncharged conductor and a point charge. In the paper referenced, ##\bar\phi_y(x)## is defined as the potential at x due to the induced charges on the conductor when the point charge is at y.
As justification for eqn 10 it states that ##\bar\phi_x(y)=\bar\phi_y(x)##.
I cannot see why that should be true, but I cannot construct any counterexample.
Is it evident, or maybe some standard result?
 
Physics news on Phys.org
Thank you for trying to clarify this point, I had already given up hope. But even if ##\bar\phi_x(y)=\bar\phi_y(x)## were true (assume this to be true for a moment, though it remains to be verified), what obvious relationship prompts the assertion that, since ##bar\phi_x(y)=\bar\phi_y(x)##, then ##\frac{q}{2} \nabla_x \phi_x(x) =- F(x)##?
 
haruspex said:
As justification for eqn 10 it states that ##\bar\phi_x(y)=\bar\phi_y(x)##.
I cannot see why that should be true, but I cannot construct any counterexample.
Is it evident, or maybe some standard result?
This is an interesting result that can be deduced from Green's Reciprocity Theorem (see equation 5).
 
  • Like
Likes PhDeezNutz, vanhees71 and haruspex
TSny said:
This is an interesting result that can be deduced from Green's Reciprocity Theorem (see equation 5).
Darn, you just beat me to it! But I'll go ahead and post my response:
Start from two charge densities ##\rho_x,\rho_y## that give rise to two electrostatic potentials ##\phi_x,\phi_y## via Poisson's equation:$$\nabla^{2}\phi_{x}\left(z\right)=-\rho_{x}\left(z\right)/\varepsilon,\;\nabla^{2}\phi_{y}\left(z\right)=-\rho_{y}\left(z\right)/\varepsilon$$and suppose that the charge densities and fields vanish sufficiently fast at spatial infinity so that boundary terms are ignorable (in other words, the usual physicist's assumption!). Then Green's second identity (https://en.wikipedia.org/wiki/Green's_identities) can be written in terms of ##\phi_1,\phi_2## as:$$0=\int\left(\phi_{x}\left(z\right)\nabla^{2}\phi_{y}\left(z\right)-\phi_{y}\left(z\right)\nabla^{2}\phi_{x}\left(z\right)\right)d^{3}z=\varepsilon^{-1}\int\left(\phi_{y}\left(z\right)\rho_{x}\left(z\right)-\phi_{x}\left(z\right)\rho_{y}\left(z\right)\right)d^{3}z$$where the integration extends over all space. This result is known in electrostatics as Green's Reciprocity (https://en.wikipedia.org/wiki/Reciprocity_(electromagnetism)). Now specialize to point charges ##\rho_{x}\left(z\right)=q_{x}\delta^{3}\left(z-x\right),\;\rho_{y}\left(z\right)=q_{y}\delta^{3}\left(z-y\right)## located at positions ##x,y##:$$0=\varepsilon^{-1}\int\left(\phi_{y}\left(z\right)q_{x}\delta^{3}\left(z-x\right)-\phi_{x}\left(z\right)q_{y}\delta^{3}\left(z-y\right)\right)d^{3}z=\varepsilon^{-1}\left(q_{x}\phi_{y}\left(x\right)-q_{y}\phi_{x}\left(y\right)\right)$$In particular, if ##q_x=q_y## (as is apparently the case in the cited external reference), then the simple reciprocal relation ##\phi_{y}\left(x\right)=\phi_{x}\left(y\right)## indeed holds.
 
  • Like
Likes vanhees71 and TSny
renormalize said:
Now specialize to point charges ##\rho_{x}\left(z\right)=q_{x}\delta^{3}\left(z-x\right),\;\rho_{y}\left(z\right)=q_{y}\delta^{3}\left(z-y\right)## located at positions ##x,y##:$$0=\varepsilon^{-1}\int\left(\phi_{y}\left(z\right)q_{x}\delta^{3}\left(z-x\right)-\phi_{x}\left(z\right)q_{y}\delta^{3}\left(z-y\right)\right)d^{3}z=\varepsilon^{-1}\left(q_{x}\phi_{y}\left(x\right)-q_{y}\phi_{x}\left(y\right)\right)$$In particular, if ##q_x=q_y## (as is apparently the case in the cited external reference), then the simple reciprocal relation ##\phi_{y}\left(x\right)=\phi_{x}\left(y\right)## indeed holds.
OK. But, besides the point charge, there will also be induced surface charge density on the conductor. So, the integration over ##\rho## should include integration over the surface of the conductor. However, you can show this integration equals zero using the fact that the conductor's surface is an equipotential surface and the net charge on the surface is zero.
 
This is from Griffiths' Electrodynamics, 3rd edition, page 352. I am trying to calculate the divergence of the Maxwell stress tensor. The tensor is given as ##T_{ij} =\epsilon_0 (E_iE_j-\frac 1 2 \delta_{ij} E^2)+\frac 1 {\mu_0}(B_iB_j-\frac 1 2 \delta_{ij} B^2)##. To make things easier, I just want to focus on the part with the electrical field, i.e. I want to find the divergence of ##E_{ij}=E_iE_j-\frac 1 2 \delta_{ij}E^2##. In matrix form, this tensor should look like this...
Thread 'Applying the Gauss (1835) formula for force between 2 parallel DC currents'
Please can anyone either:- (1) point me to a derivation of the perpendicular force (Fy) between two very long parallel wires carrying steady currents utilising the formula of Gauss for the force F along the line r between 2 charges? Or alternatively (2) point out where I have gone wrong in my method? I am having problems with calculating the direction and magnitude of the force as expected from modern (Biot-Savart-Maxwell-Lorentz) formula. Here is my method and results so far:- This...
Back
Top