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Magnetic Vector Potential

A physical quantity which has both Magnitude and direction is Known as Vector.If the divergence of the magnetic vector is zero then there is a vector )whose curl operation gives the given magnetic Vector.such A vector is called Magnetic Vector Potential.Magnetic vector potential gives the magnetic potential of the magnetic field at a particular point and specifies its direction also. 

Understanding Magnetic Induction Units  is always challenging for me but thanks to all math help websites to help me out.

Physical Significance of Vector Potential:From the definition of Magnetic Vector PotentialB = grad xx vecawhere magnetic vector Potential may be written like electrostatic potenial.VE = 1( int rho dV /r)/4piepsiwhere E is epsilonHere J is the Current density through a surface s enclosing a volume V.since,the Biot savort's Law in terms of magnetic vector potential is much simpler,so it is very easier for us to calculatefirst the vector potential is much simpler so it is easier for us to calculate first the vector Potential (veca) and then taking then curl operation on vector 

A to find the magnetic field induction,vector B(grad x A) at any pointWhy Vector a is Called the Magnetic Vector Potential:In magnetostaticsthe divergence of magnetic field induction B is Zero.divergence of vector B="0"grad ·B=0again we know that div-curl of any vector is 0grad (grad x veca )=0comparing the above tow equationsB="grad" vecaThus veca is called the vector potentialof the magnetic field of induction BHence we can define Magnetic vectorPoisson and Laplace Equation for

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Magnetic Vector PotentialFrom the Ampere circuital Law we can definegrad x B = ? ·J ..............(i)where j is the current density and B is the magnetic flux density.Again if veca is the magnetic vector potential,we can write from the definition of vector potentialB="" grad veca ..............(ii)now from the equations (i) and (ii)grad (grad x veca )=upsilon ·Jor, grad(grad ·veca )- grad 2 veca = upsilon Jor, -grad 2 veca =upsilon Jor, grad 2 veca = -upsilon J ..........(iii)This equation (iii) is known as poisson's equation for Magnetic Vector Potential