Introduction to Vector Calculus (27)

Types of vector fields (i) Solenoidal and Irrotational field (Lamellar)   if curl R 0  R  grad  where  is the scalar potential.   div grad  = 2   0 (given div R 0 ) This equation is known as Lapalce's equation and such fields are called Laplacians. e.g. Electrostatic field in free space, gravitational field in free space, thermal fields in equilibrium, magnetostatic fields in current free region, static current field within a linear homogenous isotropic conductor. (ii) Irrotational but not solenoidal field   Here curl R 0 but div R 0

 2 again with R = grad x, x being the scalar potential but div grad x = x  0 This is called the Poisson's equation and such fields are known as poissonian. e.g. electrostatic fields in a charged medium, electrons inside a thermionic tube, gravitational force inside a mass. (iii) Solenoidal but not irrotational field   here div R 0, but curl R 0   since curl R 0 R curl  where  is the vector potential  div R = div curl  = 0 (A vector identity)  and curl R = curl   0 = grad div   2   0

if now div  = 0 then 2   0 This is similar to Poisson's equation but it is terms of a vector potential. e.g. magnetic field within a conductor carrying a steady current, Rotational motion of an incompressible fluid, time varying electromagnetic field in charge free and current free region. Neither irrotational nor solenoidal field for this   curl R  0 and div R 0  R = grad x + curl  x = scalar potential  = vector potential   div R = div grad x + div curl   0 But div curl  = 0 (28) Introduction to Vector Calculus

so div grad x 0 This is Poisson's equation  and curl R = curl grad x + curl curl  But curl grad x = 0  curl curl   0

It can be reduced to 2   0 by assuming div  = 0. This is the most general type of field. It can be decomposed into two fields - one is lamellar with x and another solenoidal with . this is known as 'Helmholz theorem' e.g. Rotational motion of a compressible liquid. Helmholz theorem states that a vector function is determinal uniquely if the values of its curl of divergence are known at all points.