Please explain this suggested solution for the field inside a charged hollow sphere

[tellmesomething]

Homework Statement

Theres a hollow sphere whose $\sigma$ = $\sigma_0 cos \theta$ where $\theta$ is the polar angle/THE angle made with the x axis. Find field inside shell.

Relevant Equations

Expression of electric field on axis due to a uniformly charged ring = $$ \frac { Q z } { 4π \epsilon (R²+ z²)^{3/2} }
$$ where z is distance of the point from the center of the ring on the axis and R us the radius of the ring.

So I tried it out by taking a patch of area da at an angle theta from the x axis and rotating it around the axis, this gives you a cone whose, locus is that of a uniformly charged ring since all of the area is at the same angle theta and the surface charge density varies with theta.

My solution :

I used integration by substitution therefore u=cos $\theta$

Now this answer matches with the required answer

But the suggested solution was :

Now im very lost in this suggested solution and I have some questions ....
From what I gathered they have taken two spheres of opposite charges and super imposed them on each other
1) How do they know that the charge is distributed like this? That half of it is negative and the other half is positive. ..

Couldn't it be some other distribution?

2) I have derived this formula for electric field in a cavity when the cavity was in a wholely positive sphere...doesnt the half negative and half positive distribution change the formula a bit?

 

[tellmesomething]

Homework Statement: Theres a hollow sphere whose $\sigma$ = $\sigma_0 cos \theta$ where $\theta$ is the polar angle/THE angle made with the x axis. Find field inside shell.
Relevant Equations: Expression of electric field on axis due to a uniformly charged ring = $$ \frac { Q z } { 4π \epsilon (R²+ z²)^{3/2} }
$$ where z is distance of the point from the center of the ring on the axis and R us the radius of the ring.

So I tried it out by taking a patch of area da at an angle theta from the x axis and rotating it around the axis, this gives you a cone whose, locus is that of a uniformly charged ring since all of the area is at the same angle theta and the surface charge density varies with theta.

My solution :View attachment 345940
View attachment 345941

I used integration by substitution therefore u=cos $\theta$

View attachment 345942
Now this answer matches with the required answer

But the suggested solution was :View attachment 345943

Now im very lost in this suggested solution and I have some questions ....
From what I gathered they have taken two spheres of opposite charges and super imposed them on each other
1) How do they know that the charge is distributed like this? That half of it is negative and the other half is positive. ..

Couldn't it be some other distribution?

2) I have derived this formula for electric field in a cavity when the cavity was in a wholely positive sphere...doesnt the half negative and half positive distribution change the formula a bit?

Ive understood my First doubt..its merely the angle since cos theta becomes negative after 90°, the distribution becomes negative..the second doubt still somewhat remains

 

[Orodruin]

2) I have derived this formula for electric field in a cavity when the cavity was in a wholely positive sphere...doesnt the half negative and half positive distribution change the formula a bit?

The two superimposed spheres are both uniformly charged. The field is the sum of their fields.

 

[tellmesomething]

The two superimposed spheres are both uniformly charged. The field is the sum of their fields.

Yes I figured it out. Thankyou :)

 

[Orodruin]

Is the problem to find the field at the centre or anywhere inside the sphere?

 

[tellmesomething]

Is the problem to find the field at the centre or anywhere inside the sphere?

It is technically anywhere inside the sphere since the result is independent of distance, its uniform..

 

[Orodruin]

It is technically anywhere inside the sphere since the result is independent of distance, its uniform..

It is indeed, but this would have to be proven.

 

[tellmesomething]

It is indeed, but this would have to be proven.

I dont follow? Would I have to prove it more formally than the solution above?

 

[Orodruin]

I dont follow? Would I have to prove it more formally than the solution above?

You have not shown that this is the case. It is not sufficient to just state that the field is uniform. You need to be able to argue that this is actually the case.

 

[tellmesomething]

You have not shown that this is the case. It is not sufficient to just state that the field is uniform. You need to be able to argue that this is actually the case.

Oh. I see.. Well I dont think I know how to...

 

[Orodruin]

Oh. I see.. Well I dont think I know how to...

There are several possibilities depending on the level you start at. If you are familiar with vector addition and the field inside a uniformly charged sphere then you can use that. If you are more mathematically inclined you can solve Poisson’s equation to find the actual potential inside the sphere or solve the full integral for an arbitrary point.

 

[tellmesomething]

There are several possibilities depending on the level you start at. If you are familiar with vector addition and the field inside a uniformly charged sphere then you can use that. If you are more mathematically inclined you can solve Poisson’s equation to find the actual potential inside the sphere or solve the full integral for an arbitrary point.

Im familiar with the first one. Do I just have to take any other point and prove that the field will be the same there?

 

[Orodruin]

Im familiar with the first one. Do I just have to take any other point and prove that the field will be the same there?

Just do what they did in the suggested solution. Write down the field inside each of the uniformly charged spheres, then add them up.

 

[tellmesomething]

Just do what they did in the suggested solution. Write down the field inside each of the uniformly charged spheres, then add them up.

No that I did as I said in post #4. I thought meant an additional something after even doing the above

 

[Orodruin]

No that I did as I said in post #4. I thought meant an additional something after even doing the above

So if you do the vector additions of the fields proportional to $\vec x$, then yes you get the uniformity. I am just saying it because the full argument has not been presented in the thread. It would be useful to do so both for future readers as well as for having confirmation that you got it correct.

 

[tellmesomething]

So if you do the vector additions of the fields proportional to $\vec x$, then yes you get the uniformity. I am just saying it because the full argument has not been presented in the thread. It would be useful to do so both for future readers as well as for having confirmation that you got it correct.

Oh I see. In that case I will fair ir and post here, if you have some time please check. I would have edited it into the question but sadly I cannot edit it anymore

 

[brotherbobby]

@tellmesomething - I will advise you to take a bit of time and type out your solutions on here, using $\rm{LaTeX}$. It is difficult to read from pictures of pages on which you have done your solution due to various reasons - one's handwriting and the lighting in the room being two of them.

Additionally, if you need to draw, there are excellent drawing softwares available, with Inkscape being my favourite. It's free. Inkspace even allows you to mark places in your drawing or write formulas in it using LaTeX.

The combined effect of typing your solutions along with the drawings is that it would be much easier for people to read and understand what you have been trying to do.

 

[tellmesomething]

@tellmesomething - I will advise you to take a bit of time and type out your solutions on here, using $\rm{LaTeX}$. It is difficult to read from pictures of pages on which you have done your solution due to various reasons - one's handwriting and the lighting in the room being two of them.

Additionally, if you need to draw, there are excellent drawing softwares available, with Inkscape being my favourite. It's free. Inkspace even allows you to mark places in your drawing or write formulas in it using LaTeX.

The combined effect of typing your solutions along with the drawings is that it would be much easier for people to read and understand what you have been trying to do.

Noted. Thankyou :)