(Vectors are typed as Bold )
First, let us find out the magnetic field at test point r for the spinning charged sphere.
This is done by finding the magneteic vector potential A .
Then magnetic field is obtained from vector potential using B = x A .
Left side figure shows a charged sphere of radius R, spinning with angular speed ω .
It is required to find the vector potential at test point r whose position vctor makes angle ψ with the spinning axis.
Let us asume the coordinate system as shown in right side of figure so that centre of sphere coincides with origin,
position vector of test point r aligns with z-axis and spinning axis is rotated to lie in X-Z plane
Let us consider small area element da' = R2sinθ' dθ' dφ' . Vector potential dA at test point r is given by
........................(1)
where K(r') is the surface current density, s is the ditance between area element da' and test point r.
vector potential A(r) due to whole spinning sphere, .................................(2)
we have, K(r') = σ v , and s = ( R2 + r2 - 2Rr cosθ' )
where σ is surface charge density, v is velocity vector of area element da'
we have,
It has to be noticed that each of these terms, except one, involves either sinφ' or cosφ' .
Since integration of sinφ' or cosφ' within limits 0 to 2π vanishes, with only one non-vanishing term,
vector potential relation (2) becomes
........................ (3)
Above integration can be worked out using u = cosθ' , then integration eqn.(3) becomes
.....................................(4)
Let us consider, ω × r = - ω r sinψ .
Also when test point r is outside of sphere r > R, then bracketed term in eqn.(4) becomes -2R3
Hence .................................(5)
Next step is to adjust the coordinate system, so that spinning axis coincides with z-axis and the point r is at (r, θ, φ)
Hence eqn.(5) becomes, ..............................(6)
where in eqn.(6), C = ( μo R4 ω σ )/3 .............................(7)
Now to get magnetic field, we use B = x A
In Spherical coordinate system, using curl of vector potential , magnetic field is written as,
......................................(8)
we know electric field of sphere at test point r,
E(r) = [ Q/(4πεor2) ] = [ (4πR2σ) / (4πεor2) ] = [ (R2σ) / (εor2) ] ........................(9)
Poynting vector S = (1/μo) [ E × B ] = (1/μo) [ (R2σ) / (εor2) ] ( C sinθ / r3 )
By substituting C from eqn.(7), by considering r = R very near to spherical surface, Poynting vector becomes
S(θ) = [ (σ2 ω R sinθ )/εo ]