The curl of an arbitrary vector, \vec{A} is The curl of an arbi

pavitorj6

pavitorj6

Answered question

2021-11-30

The curl of an arbitrary vector, A is The curl of an arbitrary vector A in spherical coordinates
×A1r2sinθ[r^rθ^rsinθϕ^rθϕArrAθrsinθAϕ]
=r^rsinθ[θ(Aϕsinθ)Aθϕ]+θ^rsinθ[ArϕsinθAr}{r}(rAϕ)]+ϕ^r[}{r}(rAθ)Ar}{θ}]
Can I simply let =E and A=B  to say that the cross product of E and B in spherical coordinates
E×B=r^rsinθ(EθBϕsinθEϕBϕ)+θ^rsinθ(EϕBrrsinθErBϕ)+ϕ^r(rErBθEθBr)

Answer & Explanation

Geraldine Flores

Geraldine Flores

Beginner2021-12-01Added 21 answers

The cross product in spherical coordinates is given by the rule,
ϕ^×r^=θ^
θ^×ϕ^=r^
r^×θ^=ϕ^
this would result in the determinant,
A×B=|r^θ^ϕ^ArAθAϕBrBθBϕ|
This rule can be verified by writing these unit vectors in Cartesian coordinates.
The scale factors are only present in the determinant for the curl. This has to do with the definition of the curl and its use of length and area.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?