Laplace's and Poisson's equations(Magnetostatics)

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Laplace’s equation(Magneto statics):-

From the equation $\oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0\Rightarrow \overrightarrow{\bigtriangledown } .\overrightarrow{B}=0$ .

since $\overrightarrow{B}=\mu _{O}\overrightarrow{H}$

$\overrightarrow{\bigtriangledown }.(\mu _{O}\overrightarrow{H}) =0$ .

$\mu _{O}(\overrightarrow{\bigtriangledown }.\overrightarrow{H}) =0$ .

we know that for zero current density $\overrightarrow{J}=0$ , the Magnetic scalar potential is $\overrightarrow{H}=-\overrightarrow{\bigtriangledown }V_{m}$

by replacing $\overrightarrow{H}$ in the equation $\mu _{O}(\overrightarrow{\bigtriangledown }.\overrightarrow{H}) =0$ ,

$\mu _{O}(\overrightarrow{\bigtriangledown }.(-\overrightarrow{\bigtriangledown }V_{m})) =0$ .

$\mu _{O}(\bigtriangledown^{2}V_{m}) =0$ .

$\bigtriangledown^{2}V_{m} =0$ . is known as Laplce’s equation in Magneto statics.

Poisson’s equation(Magneto statics):-

In vector algebra a vector can be fully defined if it’s curl and divergence are defined.

$\overrightarrow{\bigtriangledown } X\overrightarrow{H}=\overrightarrow{J}$ -point form of Ampere’s law

$\overrightarrow{\bigtriangledown } X\overrightarrow{B}=\mu _{o}\overrightarrow{J}$ because $\overrightarrow{B}=\mu _{O}\overrightarrow{H}$ .

from vector Magnetic potential $\overrightarrow{B}=\overrightarrow{\bigtriangledown } X \overrightarrow{A}$ .

$\overrightarrow{\bigtriangledown } X(\overrightarrow{\bigtriangledown } X \overrightarrow{A})=\mu _{o}\overrightarrow{J}$ .

from the vector identity $\bigtriangledown ^{2}\overrightarrow{A}=\overrightarrow{\bigtriangledown }(\overrightarrow{\bigtriangledown }.\overrightarrow{A})-\overrightarrow{\bigtriangledown } X\overrightarrow{\bigtriangledown } X \overrightarrow{A}$ .

$\overrightarrow{\bigtriangledown }(\overrightarrow{\bigtriangledown }.\overrightarrow{A})-\bigtriangledown ^{2}\overrightarrow{A}=\mu _{o}\overrightarrow{J}$ .

if $\overrightarrow{\bigtriangledown }.\overrightarrow{A}=0$ .

$\bigtriangledown ^{2}\overrightarrow{A}=-\mu _{o}\overrightarrow{J}$ – this equation is known as Poisson’s equation for Magneto statics.

$\overrightarrow{A}=A_{x}\overrightarrow{a_{x}}+A_{y}\overrightarrow{a_{y}}+A_{z}\overrightarrow{a_{z}}$ and $\overrightarrow{J}=J_{x}\overrightarrow{a_{x}}+J_{y}\overrightarrow{a_{y}}+J_{z}\overrightarrow{a_{z}}$

$\bigtriangledown ^{2}(A_{x}\overrightarrow{a_{x}}+A_{y}\overrightarrow{a_{y}}+A_{z}\overrightarrow{a_{z}})=-\mu _{o}(J_{x}\overrightarrow{a_{x}}+J_{y}\overrightarrow{a_{y}}+J_{z}\overrightarrow{a_{z}})$ .

by equating the respective components on each side

$\bigtriangledown ^{2}A_{x}=-\mu _{o}J_{x}$ , $\bigtriangledown ^{2}A_{y}=-\mu _{o}J_{y}$ and $\bigtriangledown ^{2}A_{z}=-\mu _{o}J_{z}$ are the scalar Poisson’s equations of Magneto statics.