Laplace’s and Poisson’s equations(Magnetostatics)

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.

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.