How to calculate the electric potential inside of a sphere

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Homework Help Overview

The problem involves calculating the electric potential inside a sphere with a given charge density, expressed as ρ = a*r + b/r. The outer potential is defined as φ = 0 at the sphere's radius of 58.4 m, and the task is to find the potential at a distance of 15.4 m from the center.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the integration of the electric field to find the potential, with some questioning the correctness of the electric field calculation. There are mentions of needing to perform two integrations to determine the electric field and potential accurately. One participant suggests an alternative method involving the Poisson equation.

Discussion Status

The discussion is ongoing, with participants providing insights and questioning the assumptions made in the original poster's approach. Some guidance has been offered regarding the need for correct integration and the potential use of alternative methods, but no consensus has been reached on a specific solution.

Contextual Notes

There is a lack of clarity regarding the variable R in the equations presented, and participants are exploring the implications of the charge density function on the calculations. The original poster expresses confusion about the integration process and the parameters involved.

Vosegus
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Homework Statement


The density of the charge inside of the sphere is given-
ro=a*r+b/r
The electric potential on the outer layer of the sphere is phi=0
The radius of the sphere is 58.4m
r-is the distance from the center of the sphere
What is the electric potential when r=15.4m?

Homework Equations

The Attempt at a Solution


Basically, what I tried to do the to do the field equation and integrate it between 58.4 and 15.4.
That didnt work so I am clueless to what I need to do.
4*pi*r^2*E=(4*pi*R^3)ro/3epsilon
 
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Like potential, the Electric field also varies with the distance from center. Did you get your electric field right? Your integration parameters seem to be fine, it may be possible that your Electric field is not right.
 
Last edited:
Vosegus said:

Homework Statement


The density of the charge inside of the sphere is given-
ro=a*r+b/r
The electric potential on the outer layer of the sphere is phi=0
The radius of the sphere is 58.4m
r-is the distance from the center of the sphere
What is the electric potential when r=15.4m?

Homework Equations

The Attempt at a Solution


Basically, what I tried to do the to do the field equation and integrate it between 58.4 and 15.4.
That didn't work so I'm clueless to what I need to do.
4*pi*r^2*E=(4*pi*R^3)ro/3epsilon
What is R? This equation makes no sense.
You have to do two integrations:
the first is to get the charge within a given r to get E(r), along the lines of your equation but fixed up;
then you integrate E(r) from the outer surface to r = 15.4m.
Watch your signs.
 
Just FYI this problem can also be solved by solving the Poisson equation. In this case it becomes the second-order Euler-Cauchy equation. Boundary conditions are the potential and the E field at the surface. Just thought I'd throw it out in case you go on to advanced methods (actually not that advanced, let's call them "alternative approaches" though the math is more demanding than what you need to solve this problem your way.
 

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