Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 65

ˆ

ˆ

ˆ

P

T B

R

B

Figure 2.13 Local coordinate system used to describe a curved magnetic field. The inset
gives the directions of the unit vectors at the point P . In a stretched magnetic field the
tension force T B is parallel to and inversely proportional to R .

b

normal to B (and antiparallel to the radius of curvature), and a unit vector

given by b

=ˆ n ׈ s . We have just shown that J × B may be written in the form

of the magnetic force f B = J × B ,or

B 2

μ 0 f B = ( 1 /μ 0 )( B ·∇ ) B −∇ / 2 (2.61)

where

B = B s ˆ

n , and b ,

In terms of ˆ s , ˆ

B 2 ( B ·∇ ) B = B (∂/∂ s )( B ˆ s ) = B (∂ B /∂ s ) ˆ s + (∂ ˆ s /∂ s )

Now

∂ ˆ s /∂ s =ˆ nd θ/ Rd θ =ˆ n / R

so

B 2

R ( B ·∇ ) B = B (∂ B /∂ s ) ˆ s + / n ˆ

Substituting into (2.61),

B 2

μ 0 R

B 2

μ 0

f B = / 2 n ˆ + ( B /μ 0 )(∂ B /∂ s ) ˆ s −∇ / 2

But

B 2

μ 0

( 1 /μ 0 )( B ∂ B /∂ s ) = ∂ / 2 /∂ s