v dt

dl dA

v dt

t = dt

dl

t=0

Figure 4: The area dA covered in time dt by a line element dl of a curve which is moving with the flow speed v. The thin arrows mark the displacement v dt.

# Adding the contributions from (I) and (II) (equations (63) and (65), we get

dΦ dt

=

dΦ dt

I

+

dΦ dt

II

=−

C

(E + v × B) · dl.

(66)

But according to (59), the integrand will be zero for slow and large-scale phenomena. Therefore, for such phenomena, the flux through any closed curve following the plasma flow will be constant. This is known as the “freezing-in” of the magnetic field into the plasma.

It is possible to show that the flux conservation implies that two elements of plasma which at one time are on the same magnetic field line always will be so. It is therefore possible to picture the magnetic field lines as ropes frozen into the plasma and following its motion.

# 4.6 Energy densities in a plasma

We can write the total energy W in a volume V of plasma as the integral of an energy density w, W = _{V }w dV . The energy has contributions from four sources. Most obvious is perhaps the

thermal energy density due to the random motion of ions and electrons,

w_{T }

=

3 2

n i K T i +

3 2

n e K T e

.

(67)

We also have a kinetic energy density due to the ordered motion (flow) of the plasma:

w_{K }

=

1

2

n i m i v 2 i +

1

2

n i m i v 2 i .

(68)

These are the terms that would appear in an ordinary neutral gas. However, in a plasma one also needs to consider the energy contained in the electromagnetic fields, since this energy can be con- verted to particle energy and vice versa. Thus, we have to include the energy density due to the

existence of a magnetic field: 1

w_{B }

=

2µ_{0 }

B^{2}.

(69)

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