Electric field in a solid cylinder with an offset hole

[calandra]

[SOLVED] Electric field in a solid cylinder with an offset hole

Homework Statement

its teenie, but it says: Superposition: an infinite cylinder of radius R and uniform charge density (row) contains a cylindrical cavity of radius R/2 as shown. Find the electric field in the cavity.

Homework Equations

Gauss's Law I guess.

The Attempt at a Solution

My first thought was 'zero. duh'...but that's not exactly superposition. I know there has to be something about the whole cylinder minus the hole...but I'm not sure how to go about integrating with a common origin or...I've just gotten myself very confused. HELP!

 

[Doc Al]

Here's a hint: A negative charge can cancel a positive charge.

 

[Rainbow Child]

Let $E_1$ the wanted electric field, $E_2$ the electric field produced by the cylinder $R/2$ alone and $E_3$ the electric field produced by the cylinder $R$. Then

$$E_1+E_2=E_3\Rightarrow E_1=E_3-E_2$$


The electric fields $E_2, E_3$ are easily obtained by Gauss law.

 

[calandra]

thank you thank you
however...easily obtained by gauss's law...easily...I'm having a very blonde afternoon...I can't seem to pick a reference frame that works for both E1 and E2. And do I treat E2 as a cylindrical shell with uniform charge distribution? Or as nothing...not that it really matters...its still zero right?

 

[Doc Al]

Come up with two solid uniform cylinders of charge which add up to give you the needed charge configuration.

 

[MrButterbean]

Hey there, I take it you're in Phys 231/239 as well?

 

[calandra]

wow...lol.
Yup 239.
Have you finished it?

 

[MrButterbean]

I'm done the assignment except for this question. I too was having some trouble combining the two fields (awkward using two separate cylindrical coordinate systems). I think I've figured it out now though.

I'm technically in 231, but it's just 239 renamed for us purple folk.

 

[irish_coffee]

Come up with two solid uniform cylinders of charge which add up to give you the needed charge configuration.

Hi, I'm having trouble with the same problem. The center of the cavity is at a distance A from the center of the big cylinder. My calculations of superpositioning the two fields yields an electric field within the cavity equal to zero. Is this correct?

 

[Doc Al]

My calculations of superpositioning the two fields yields an electric field within the cavity equal to zero. Is this correct?

No, that's not correct. Show how you came to that conclusion.

 

[irish_coffee]

I know, I discovered it was nonsense. I now have calculated the vector product of the vector field produced by the big solid cylinder with charge density rho and the vector field produced by the small cylinder with charge density minus rho. Because this adds up to the situation we have here (a big cylinder with a cavity with density of zero). The direction of the resulting vector field in an arbitrary point P within the cavity is in the direction of the vector connecting the center of the big cylinder and the center of the cavity.