Cylindrical symmetry, Gauss's Law

[mawgs]

Homework Statement

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A semiconducting nanowire has a volume charge density ρ(r)=ρ0(r/R) where R is the radius of the nanowire. How would you calculate the electric field inside the wire?

Homework Equations

Gauss's Law

The Attempt at a Solution

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I know that by symmetry the E field only points radially out. Using Gauss's law and finding the dq by integrating in terms of dr, a small ring in the gaussian cylinder. So would you integrate ρ0(r/R)2piLdr from 0 to r?

 

[haruspex]

would you integrate ρ0(r/R)2piLdr from 0 to r?

That is a reasonable integral to perform, but what exactly do you think it will give you?

 

[mawgs]

Qenc?

 

[haruspex]

Qenc?

Yes. So what is the next step to find the field?

 

[mawgs]

Yes. So what is the next step to find the field?

Set the Qenc over E0 equal to E2πrL. When I did this, I got ρ0r/E0R. I know this is wrong because the homework is online and gives instant feedback. When I integrated, I got .5(2πρ0Lr)/R. I'm not sure where I went wrong.

 

[haruspex]

Set the Qenc over E0 equal to E2πrL.

Sorry, what's E0? Sounds like a field. Why would you divide the charge by a field?

I know this is wrong

You haven't taken into account the charge between r and R.

 

[mawgs]

Sorry, what's E0? Sounds like a field. Why would you divide the charge by a field?

You haven't taken into account the charge between r and R.

Sorry, I meant epsilon naught.

 

[haruspex]

Sorry, I meant epsilon naught.

Ok. But I think you lost a factor of 2 in there somewhere.