Use of imaginary charge vs Gauss's 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.