Use of imaginary charge vs Gauss' law

[kirito]

Homework Statement

Consider a spherical conducting shell with a total charge of
−q uniformly distributed over its surface. Inside this shell, there is a point charge
+q placed at a point on the z-axis, at a distance less than the radius of the spherical shell.
Find the electric field in all of space (both inside and outside the spherical shell).

Relevant Equations

Imaginary charge , gauss, Laplace equation either can be used

I tried to solve the question using two different approaches to gain a better understanding of the subject. However, I reached two different results with each approach.

I believe I used Gauss's law to find the electric charge distribution and the electric field inside the cavity incorrectly .since I assumed that the field is constant and took the electric field outside the integral. But when I visualized the field around a charge displaced from the center, it started to behave oddly—being zero at some locations and non-zero at others especially if viewed from a reference frame such that the center of the sphere is the origin point .

I think this is why I obtained incorrect results.
That said, I do remember successfully solving several problems using Gauss's law that had both conductor cavity and charge , but I believe those were in electrostatic equilibrium conditions.

 

[PhDeezNutz]

They are both electrostatic equilibrium conditions.

Gauss’ Law can only be used in cases of symmetry. Planar, Cylindrical, or Spherical.

In your case do you really expect a point charge off center from the center of a sphere to not produce a field whose direction and magnitude are constant over said surface?

 

[kirito]

They are both electrostatic equilibrium conditions.

Gauss’ Law can only be used in cases of symmetry. Planar, Cylindrical, or Spherical.

In your case do you really expect a point charge off center from the center of a sphere to not produce a field whose direction and magnitude are constant over said surface?

I do not

 

[PhDeezNutz]

I do not

Then you can’t use Gauss’ Law.

Here is a good article for you to read if you want more clarification.

https://www.physicsforums.com/insights/a-physics-misconception-with-gauss-law/

 

[kirito]

Then you can’t use Gauss’ Law.

Here is a good article for you to read if you want more clarification.

https://www.physicsforums.com/insights/a-physics-misconception-with-gauss-law/

Thank you! It provided clarity and reminded me to always recheck the conditions of any simplification, rather than becoming too familiar with the result and assume the conditions apply intuitively.i overlooked the issues until something seemed off with the solution.

 

[PhDeezNutz]

Thank you! It provided clarity and reminded me to always recheck the conditions of any simplification, rather than becoming too familiar with the result and assume the conditions apply intuitively.i overlooked the issues until something seemed off with the solution.

Great to hear!! Gotta give kudos to @Orodruin the author of that insights article.

 

[Orodruin]

Gauss’ Law can only be used in cases of symmetry. Planar, Cylindrical, or Spherical.

Then you can’t use Gauss’ Law.

Great to hear!! Gotta give kudos to @Orodruin the author of that insights article.

I’d like to clarify this: Gauss law is still perfectly valid. You just can’t use it to easily find the electric field.

 

[kuruman]

You just can’t use it to easily find the electric field.

And that's because

I assumed that the field is constant and took the electric field outside the integral

this assumption was unjustified. You can take the integral form of Gauss's law $$\int_S\mathbf E\cdot \mathbf{\hat n}~dS =\frac{q_{\text{enc.}}}{\epsilon_0}$$ one step further and divide both sides by the surface area $~S=\int dS~$ over which you integrate to get $$\frac{\int_S\mathbf E\cdot \mathbf{\hat n}~dS}{\int dS} =\frac{q_{\text{enc.}}}{\epsilon_0 ~S}.$$The left hand side is the surface-averaged normal component of the electric field, $$\langle E_{\perp}\rangle_s=\frac{q_{\text{enc.}}}{\epsilon_0 ~S}.$$This last expression says it all. You can move the charge to anywhere you want inside the shell and the average value of the normal component will not change. However the normal component at the surface is equal to the average only when it has the same value everywhere on the surface.

Note that you can also squash and stretch the surface into a blob of some kind. As long as you don't change the total area $S$ and the enclosed charge $q_{\text{enc.}}$, the average normal component of the electric field on the surface will not change.

 

[Orodruin]

And that's because

this assumption was unjustified. You can take the integral form of Gauss's law $$\int_S\mathbf E\cdot \mathbf{\hat n}~dS =\frac{q_{\text{enc.}}}{\epsilon_0}$$ one step further and divide both sides by the surface area $~S=\int dS~$ over which you integrate to get $$\frac{\int_S\mathbf E\cdot \mathbf{\hat n}~dS}{\int dS} =\frac{q_{\text{enc.}}}{\epsilon_0 ~S}.$$The left hand side is the surface-averaged normal component of the electric field, $$\langle E_{\perp}\rangle_s=\frac{q_{\text{enc.}}}{\epsilon_0 ~S}.$$This last expression says it all. You can move the charge to anywhere you want inside the shell and the average value of the normal component will not change. However the normal component at the surface is equal to the average only when it has the same value everywhere on the surface.

Note that you can also squash and stretch the surface into a blob of some kind. As long as you don't change the total area $S$ and the enclosed charge $q_{\text{enc.}}$, the average normal component of the electric field on the surface will not change.

People doing this all the time was the entire reason I wrote the insight @PhDeezNutz linked to earlier …

 

[kuruman]

People doing this all the time was the entire reason I wrote the insight @PhDeezNutz linked to earlier …

Yes, and when these people are corrected, they draw the wrong inference that Gauss's law is only valid in cases of high symmetry. I think that instructors need to teach Gauss's law taking these misconceptions under consideration and not jump head first to finding electric fields in cases of high symmetry.

When I taught algebra-based physics I introduced an experimental investigation of Gauss's law in 2D using graphite paper on which closed loop "Gaussian" surfaces were drawn with conducting ink. Instead of mapping equipotentials, students measured the potential difference across the prongs of a probe in a direction perpendicular to the loop and added the measurements over the closed loop. Anyone who is interested in hearing more, just send me a PM.

 

[PhDeezNutz]

I should have been more clear in my posts.....The electric field flux through a surface is ALWAYS proportional to the charge enclosed (regardless of symmetry) but it's NOT ALWAYS USEFUL in finding the field (for that you need a high degree of symmetry).

 

[kuruman]

I should have been more clear in my posts.....The electric field flux through a surface is ALWAYS proportional to the charge enclosed (regardless of symmetry) but it's NOT ALWAYS USEFUL in finding the field (for that you need a high degree of symmetry).

Needs to be clarified by adding two words
The electric field flux through a closed surface is ALWAYS proportional to the total charge enclosed (regardless of symmetry) but . . .

 

[PhDeezNutz]

Needs to be clarified by adding two words
The electric field flux through a closed surface is ALWAYS proportional to the total charge enclosed (regardless of symmetry) but . . .

That’s fair.