Running head: ELECTRIC FIELD OF A CHARGED SPHERE1Electric Field of a Charged SphereNameInstitution
ELECTRIC FIELD OF A CHARGED SPHERE2Minimal Principles of Energy in ElectrostaticsIntroductionElectrostatics 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 MethodMinimizing 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 12ε°¿ over the volume. Therefore energy is calculated as follows:F=12ε°∫01¿¿ --------------- (i)But since e = 0, the second term becomes zero at the boundary conditions. Therefore,∫01(dψdx)dedxdx=¿∫01[ddx(edψdx)−ed²ψdx²]dx=edψdx|10+∫01e.0dx=0 ------------------ (ii)But the Laplace equation d²ψdx²=0 therefore if we substitute equation (ii) into equation (i) we come up with,
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