Report: Electric Field Analysis of a Charged Sphere

This report analyzes the electric field of a charged sphere, focusing on electrostatics principles and numerical methods. It utilizes COMSOL Multiphysics to explain Poisson's equation and explores the finite difference method and the finite element method for solving the Poisson's and Laplace equations. The report details the minimal principles of energy in electrostatics, deriving equations for energy calculation and demonstrating how the finite element method is applied to approximate the Poisson's equation. The report also provides the boundary conditions and the approximation of the Poisson's equation in a one-dimensional system using polynomial functions, with relevant references to support the analysis. The report concludes by solving the equation representing the minimum energy and taking into account the assumption that there is no space charge.

Running head: ELECTRIC FIELD OF A CHARGED SPHERE 1
Electric Field of a Charged Sphere
Name
Institution

ELECTRIC FIELD OF A CHARGED SPHERE 2
Minimal Principles of Energy in Electrostatics
Introduction
Electrostatics is the study of static charges. The principle of electrostatics is well described by
the Coulomb’s law. In the evaluation of the electric field around a charged sphere, one uses
COMSOL Multiphysics. This package helps in explaining partial differential equations like the
Poisson’s ratio (Comsol Multiphysics, 2012). Numerical methods that help in solving the
Poisson’s equation include the finite difference method and the finite element method.
The Finite Difference Method
Minimizing the total energy in the solution’s area satisfies both the Poisson’s and Laplace
equations. We need to focus on the one-dimensional case (Dhatt et al., 2012). Taking the
example of Laplace equation, in the nonexistence of electric charge, the energy of the system is
evaluated by integrating 1
2 ε° ¿ over the volume. Therefore energy is calculated as follows:
F= 1
2 ε° ∫
0
1
¿ ¿ --------------- (i)
But since e = 0, the second term becomes zero at the boundary conditions. Therefore,
∫
0
1
( dψ
dx ) de
dx dx=¿ ∫
0
1
[ d
dx (e dψ
dx )−e d ² ψ
dx ² ]dx =e dψ
dx | 1
0+∫
0
1
e .0 dx=0 ------------------ (ii)
But the Laplace equation d ²ψ
dx ² =0 therefore if we substitute equation (ii) into equation (i) we
come up with,

ELECTRIC FIELD OF A CHARGED SPHERE 3
F= 1
2 ε° ∫
0
1
¿ ¿ ------------- (iii)
The first term on the right-hand side stands for the total energy of the system and the electric
distribution of u(x) resembles that of ψ(x) and thus according to the uniqueness theorem,
u ( x )=¿ψ(x) if e(x) = 0
Thus from the above equations, it is evident that the solutions to the Poisson’s or Laplace
equations for an electric field distributed spatially matches with the least field energy integrated
over the volume of the system.
The Finite Element Method
To come up with the best estimation of Poisson’s ratio, one needs to use the finite element
method. In this method there are three very important steps:
 Find the parameters of u(x)
 Regarding the parameters, derive the equation for the total energy of the system
 Evaluate the parameter values that will make the system’s energy minimum (Hughes,
2012)
Since our focus is on a one-dimensional system, we find the parameters by dividing the space
into small intervals known as finite elements. We then approximate the Poisson’s equation for
each of the elements by a polynomial function, most probably a straight line or a parabola. For a
one-dimensional system, we use the following boundary conditions:
ψ=0 at x=0∧ψ=1 at x=1 (James R. Nagel, 2012)

ELECTRIC FIELD OF A CHARGED SPHERE 4
We conclude by assuming that the space charge = 0 and the furthermore, we estimate the answer
to the Poisson’s equation in each of the elements by a polynomial of the first order. Dividing the
space between zero and one into N finite elements and doing an approximation of ψ by a linear
function u. We therefore, come up with;
u=b1 x +a1 for 0<x < x1 ------------ (iv)u=b2 x+a2 for x1< x< x2 ------------ (v)
u=bn x+an for xn−1< x<xn ----------- (vi)
To find the equation representing the minimum energy, we solve the following expression,
F=∫
0
x1
b1
2 dx +∫
x1
x2
b2
2 dx +…+ ∫
xn−1
1
bn
2 dx=b1
2 x1 +b2
2
( x2−x1 ) +…+ bn
2
( 1−xn−1 ) --(vii)
F= u1
2
x1
+¿ ¿ -------- (viii)
Minimizing F leads to,
∂ F
∂U n−1
=0❑
⇔ 2 U n−1−Un−2
Xn−1−Xn−2
−2 1−U n−1
1−Xn−1
=0 --------------- (ix)
The above equations take into account the assumption that there is no space charge.

ELECTRIC FIELD OF A CHARGED SPHERE 5
References
Comsol Multiphysics, reference guide, may 2012,
http://hpc.mtech.edu/comsol/pdf/mph/COMSOLMultiphysicsReferenceGuide.pdf
Dhatt, G., LefranÃ, E., & Touzot, G. (2012). Finite element method. John Wiley & Sons.
Hughes, T. J. (2012). The finite element method: linear static and dynamic finite element
analysis. Courier Corporation.
James R. Nagel, Introduction to the Finite Element Method, Department of Electrical and
Computer Engineering, University of Utah, Salt Lake City, Utah, 2012:
http://www.ece.utah.edu/~ece6340/LECTURES/Apr2/Nagel_FEM.pdf