- #1

timetraveller123

- 621

- 45

## Homework Statement

ok this is not a homework but i am confused by the derivation of ## f = \nabla (m . B)##

both start out with ##\vec B(r) \approx \vec B_0 + (\vec r' . \nabla)\vec B|_0 ... \\

f = \iiint_{v'} \vec J( r') \times (\vec r' . \nabla)\vec B|_0

##

where prime is body coordinates and unprime is space coordinates

the first one uses vector identities

##

(\vec r' . \nabla)\vec B = \nabla (\vec r' . B) - r' \times (\nabla \times B)

##

the other two terms die out because of different coordinates

then it proceeds to say curl of magnetic field is zero about the origin if it is due to external sources

then the proof proceeds i have posted in the photo

the second proof(from girffiths) uses le cevitas and kronecker deltas

while both seem to arrive at the same answer of ## f = \nabla(\vec m . B)## after this griffiths specifically mentions that

he mentions that ##\nabla \times B \neq 0##

honestly to me the first proof makes more sense because the magnetic field used is due to external sources and should not have curl at the origin but questioning seems griffiths seems inplausible so

what actually is correct sorry for the long post it has been bugging me for a long while now