Electric Potential by Integrating Poisson's Equation

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SUMMARY

The discussion focuses on deriving the electric potential \(\varphi(r)\) from a spherical charge distribution with constant charge density \(\rho\) using Poisson's equation \(\nabla^2(\varphi) = -\frac{\rho}{\epsilon_0}\). The correct approach involves recognizing spherical symmetry and solving the resulting second-order differential equation. The potential inside the sphere is given by \(\varphi(r) = -\frac{\rho}{\epsilon_0}\frac{r^2}{6} + C\) for \(r \leq R\) and \(\varphi(r) = -\frac{C}{r} + D\) for \(r \geq R\), where \(C\) and \(D\) are constants determined by boundary conditions.

PREREQUISITES
  • Understanding of Poisson's equation and its applications in electrostatics.
  • Familiarity with spherical coordinates and the Laplacian operator in three dimensions.
  • Knowledge of boundary value problems and continuity conditions for electric potential.
  • Ability to perform integration and solve differential equations.
NEXT STEPS
  • Study the derivation of the Laplacian in spherical coordinates for various functions.
  • Learn about boundary conditions in electrostatics and their implications for potential functions.
  • Explore the relationship between electric fields and potentials, particularly in spherical symmetry.
  • Investigate the uniqueness theorem for solutions to Poisson's equation in electrostatics.
USEFUL FOR

Students and professionals in physics, particularly those focusing on electromagnetism, electrostatics, and mathematical methods in physics. This discussion is beneficial for anyone looking to deepen their understanding of electric potential derivation from charge distributions.

  • #31
Hi
I need some help with understanding like i have this poisson's equation from which i have to determine the potential by integrating ofcourse using the boundary condition phi=0 at x=plus or minus L/2 and that differentiation of phi with x is zero at x=0.

I see that they have arrived at the equation
phi= 1/2 rho/epsilon* (square of L/2 - square of x)

I don't know how they got this.

Note: How can I type physics symbols here?

Thanks
 
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  • #32
appsci said:
Hi
I need some help with understanding like i have this poisson's equation from which i have to determine the potential by integrating ofcourse using the boundary condition phi=0 at x=plus or minus L/2 and that differentiation of phi with x is zero at x=0.

I see that they have arrived at the equation
phi= 1/2 rho/epsilon* (square of L/2 - square of x)

I don't know how they got this.

Note: How can I type physics symbols here?

Thanks

Please create a new thread for your problem, and follow the homework template. Make sure you post the entire problem just like it is asked in your assignment.
 
  • #33
gabbagabbahey said:
Please create a new thread for your problem, and follow the homework template. Make sure you post the entire problem just like it is asked in your assignment.

Thanks. I will do so but can I know how to type equations or insert symbols over there.

Regards
 
  • #34
There is an introduction to using \LaTeX in these forums in this thread
 

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