[ElectroStatics]Question Regarding Charge within a Metal Sphere

[MysteryMan]

Homework Statement

I have this question on a practice exam in preparation fora final exam, and I am questioning my solution here:

A hollow spherical metal shell has an outer radius equal to 1.5 m and a
shell thickness of 0.5 m. A +100 nC point charge is located in the hollow
0.5 m from the centre. The metal has no net charge and is isolated from
ground.

a) Indicate with a clear drawing where charge density occurs on the
surfaces or inside the metal. Indicate where the charge density is larger or
smaller by using greater or lesser numbers of + or -
signs.

b) Determine the the electric field vector 2 m from the centre in the same
direction as the charge.

c) What is the total electric flux through a sphere 10 m in radius that
encloses the metal sphere?

Homework Equations

flux=charge enclosed
E=Q/(4*π*ε0*r^2)

The Attempt at a Solution

a)
The outside of the inner cavity with the sphere will have a negative charge induced, with the density being greater at the portion closer to the point charge. I also said that since the net Electric field within the metal must be zero, the positive charge density on the outside must be uniform, as it is unaffected by the inner field. This is the part I am not sure of. I also concluded that the charge on the outside must be the same 100nC of the point charge inside, effectively making the sphere a mirror of the inner point charge.

b)
Assuming the logic in (a) was sound, I asserted that the electric field vector on the point 2m away from the centre must depend on the distance from the centre of the "point charge" sphere, which is 2m as stated. So using the formula I provided above (from a formula sheet we are provided) I computed the E field to be E=224.6888 N/C (or V/m) in the positive x direction.

c)
In our class, we define the flux to be equivalent to the charge enclosed. Which would clearly be 100nC.


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Am I on the right track? Thank you very much in advance.

EDIT: I attached a screenshot of the question.

 

[Simon Bridge]

Well done. See the following p74
http://ecee.colorado.edu/~ecen3400/Chapter%206%20-%20Conductors%20in%20the%20Electrostatic%20Field.pdf

(a) is correct.

(b) Anyone outside the shell sees an image of the point charge somewhat displaced from it's actual position - in this case the charge appears to be at the center of the sphere.

(c) You'd draw a gaussian surface - and this is correct: net flux is proportional to the charge enclosed.

 

[MysteryMan]

Alright thanks for the confirmation.

 

[Simon Bridge]

(b) is related to optics ... if it were a glass sphere and the charge were an object, the object would appear, from outside, to be displaced from it's actual position. Not too surprising since light is a form of electromagnetism.

Notice, in the link, though that he field lines inside the sphere are curved?
The field lines at a conducting surface must be normal to that surface.

So I only had very small notes for you.

 

[HalfLostAndSearching]

I would say that your solution is generally correct, but there are a few points that could use further explanation or clarification.

a) Your reasoning for the charge density on the outside of the inner cavity is correct - a negative induced charge will be present due to the positive charge inside. However, it is important to note that the charge density will not be uniform on the outside surface. The density will be greater closer to the point charge, but it will decrease as you move further away from the point charge. This is because the electric field strength decreases with distance, so the induced charge will also decrease.

b) Your calculation of the electric field strength at a point 2m from the center is correct, assuming a point charge of +100 nC at the center. However, it is important to clarify that this electric field is only due to the point charge and does not take into account the induced charges on the metal sphere. If you were to consider the electric field due to both the point charge and the induced charges, it would be slightly different.

c) Your definition of flux as equivalent to the charge enclosed is correct, but it is important to note that the flux through a closed surface is actually the net electric flux, taking into account both positive and negative charges. In this case, since there are no other charges besides the +100 nC point charge, the net flux through a sphere enclosing the metal sphere would indeed be 100 nC.

Overall, your reasoning and understanding of the concepts is good. It would be helpful to provide more clarification and explanation in your solution, especially when it comes to the non-uniform charge density and the electric field due to both the point charge and induced charges.