Start with the differential form of Gauss's law:
∇ â— E = Ï/ε0, the divergence of the electric field is equal to the total charge density divided by the permittivity of free space.
Make the following substitution, assuming electrostatic charges:
E = -∇φ, the electric field at a point is equal to the negative gradient of the scalar electric potential.
This gives:
∇ ◠∇φ = -Ï/ε0
From an identity of vector calculus we get the following:
∇2φ = -Ï/ε0, which is Poisson's equation with f = -Ï/ε0.
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Considering Maxwell equations and contitutive relations. See pag.18 of principles of nano-optics, Lucas Novotny.
derive clausious mossotti equation
Independence:The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.Consistency:The equations of a linear system are consistent if they possess a common solution, and inconsistent otherwise. When the equations are inconsistent, it is possible to derive a contradiction from the equations, such as the statement that 0 = 1.Homogeneous:If the linear equations in a given system have a value of zero for all of their constant terms, the system is homogeneous.If one or more of the system's constant terms aren't zero, then the system is nonhomogeneous.
This involves the rate of change of the unit tangent vector. Deriving the curvature starts with the equation of a circle. Then three equations that force the collocation circle to go through the three points and on the curve must be written down. Then solve for a, b, and r.
It is not possible to reproduce the equations on this website, however you can find a detailed derivation at the related link.