Finding Electric Field of Charged Dielectric Sphere

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Homework Help Overview

The discussion revolves around finding the electric field of a charged dielectric sphere with a non-uniform charge density given by ρ = 3Qcos(θ)/4πR³. The sphere is centered at the origin in a Cartesian coordinate system, and the participants are tasked with determining the electric field at a point outside the sphere.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of Gauss's Law, noting that the lack of symmetry due to the charge density's dependence on θ complicates its use. Some suggest that direct integration may be necessary to find the electric field or potential.
  • There are attempts to integrate the charge density and calculate the electric potential, with questions raised about how to derive the electric field from the potential and how to express it in Cartesian coordinates.
  • Concerns are expressed about the implications of integrating the charge density leading to zero results and how that affects the electric field and potential.

Discussion Status

The discussion is ongoing, with participants exploring different methods to approach the problem. Some have provided insights into the challenges of integrating the charge density and deriving the electric field from the potential. There is no explicit consensus, but various interpretations and methods are being examined.

Contextual Notes

Participants note that the charge density leads to a net charge of zero when integrated, raising questions about the implications for the electric field and potential. There are also references to homework rules regarding multiple postings, indicating a structured forum environment.

khfrekek1992
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Homework Statement



There is a charged dielectric sphere of radius R. I center it at the origin of an x,y,z cartesian system. It has a charge density rho=3Qcos(theta)/4*Pi*R^3. Q and R are given.

I need to find the Electric field at a point that is outside the sphere.

Homework Equations



int(E*dA)=Q_in/Epsilon0


The Attempt at a Solution



int(E*dA)=Q_in/Epsilon0
E(4*Pi*r^2)=(rho)(4/3)*Pi*r^3)/epsilon0
E=3Qcos(theta)r/12*Pi*epsilon0*R^3 in the r-hat direction

Is this right? Dont I have to figure out the field in the x-hat, y-hat, and z-hat directions? How does this work if my E is in the r-hat direction?

Thanks in advance!
 
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It appears that you are trying to use Gauss's Law to find the electric field. The charge density depends on θ, so there isn't enough symmetry for Gauss's Law to be of help here.

It looks like you will have to use direct integration to find either electric field, E, or electric potential, V.
 


Thanks for the reply. I tried integrating rho directly earlier, but always got 0 because sin evaluated from 0 to Pi is always 0. :(
 
Find the Electric field at a point outside a charged sphere

Homework Statement



There is a charged dielectric sphere of radius R. I center it at the origin of an x,y,z cartesian system. It has a charge density rho=3Qcos(theta)/4*Pi*R^3. Q and R are given.

I need to find the Electric field at a point that is outside the sphere.


Homework Equations



E=-(grad)V
V=kq/r

The Attempt at a Solution



I integrated rho over the volume of the sphere to get q=(Pi/2)Qcos(theta)
then used V=q/4*pi*epsilon0r to get v=(Qcos(theta))/(8epsilon0*r)

How do i split this voltage into x,y, and z components so i can take the gradient to get E at a certain (x,y,z) point?

Thanks in advance
 


Yeah haha it is the same but I am doing the problem completely different now(from your suggestion) and am now stuck at a different part :/
 


Yes, the net charge is indeed zero. Integrating the cosine from 0 to π does give zero.

That doesn't mean that the electric potential and/or the electric field are zero. (The same can be said for an electric dipole.)
 


Oh okay that makes sense, thank you! So I found the potential by taking kq/r, and now I am taking -(grad)V to get E. So I have E=-(d/dxx-hat+d/dyy-hat+d/dzz-hat)*(3Qcos(theta))/4*Pi*R^3Epsilon0

is that right?
 


oh wait if i take the gradient of that I get 0 in every direction..
 
  • #10


khfrekek1992 said:
Oh okay that makes sense, thank you! So I found the potential by taking kq/r, and now I am taking -(grad)V to get E. So I have E=-(d/dxx-hat+d/dyy-hat+d/dzz-hat)*(3Qcos(theta))/4*Pi*R^3Epsilon0

is that right?
How did you find V ?
 
  • #11


See below
 
Last edited:
  • #12


Actually, I guess I should Multiply the charge density by the volume of the sphere to get Q, then use that in V=kQ/R to get V=(QR^2/repsilon0)cos(theta) Where everything except for r and theta is a known constant. Then I need to take the negative gradient to get E. I just don't know how to take the x,y, and z derivatives of that voltage, because they would all be 0? :/
 
  • #13


khfrekek1992 said:
Yeah haha it is the same but I am doing the problem completely different now(from your suggestion) and am now stuck at a different part :/

Please do not multiple post. It is against the PF rules.

Two threads merged into one.
 

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