example of vector potential


If the solenoidal vector  U=U(x,y,z)  is a homogeneous function of degree λ (-2),  then it has the vector potentialMathworldPlanetmath

A=1λ+2U×r, (1)

where  r=xi+yj+zk  is the position vector.

Proof.  Using the entry nabla acting on products, we first may write

×(1λ+2U×r)=1λ+2[(r)U-(U)r-(U)r+(r)U].

In the brackets the first product is, according to Euler’s theorem on homogeneous functions, equal to λU.  The second product can be written as  Uxrx+Uyry+Uzrz, which is Uxi+Uyj+Uzk, i.e. U.  The third product is, due to the sodenoidalness, equal to  0r=0.  The last product equals to 3U (see the first formula (http://planetmath.org/PositionVector) for position vector).  Thus we get the result

×(1λ+2U×r)=1λ+2[λU-U-0+3U]=U.

This means that U has the vector potential (1).

Title example of vector potential
Canonical name ExampleOfVectorPotential
Date of creation 2013-03-22 15:42:56
Last modified on 2013-03-22 15:42:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 26B12