Explanation:
- vector r lies on z- axis
- J is tilted at angle Ψ
- Orient x-axis such that w lies in x-z plane
Given:
Vector potential
Where, K = б*v ; r* = sqrt (R^2 + r^2 -2R*r*cos(θ')) ; da' = R^2*sin(θ')*dθ'dΦ'
Solution:
- Velocity of v point a point r' in a rotating rigid body is given by:
v = w x r' =
- where a = Ψ and b' = θ' and c' = Φ'
v = R*w [-(cos Ψ *sin θ' *sin Φ') x + (cos Ψ *sin θ' *cos Φ' - sin Ψ * cos θ') y
+ (cos Ψ *sin θ' *sin Φ') z ]
- Notice that terms like sin Φ' and cos Φ' contribute to zero:
- Hence,
- Evaluate integral u = cos (b')
- From we can determine two cases when r > R and r < R
Hence,
r < R
r > R
- Reverting back to original coordinate system given in figure 5.45:
r < R
r > R
Where, b = θ and c = direction along Φ.
Hence, A ( r , θ , Φ )