**:**

*Introduction*In general, Maxwell’s equations will be expressed in vector form:

i)

**Ñ****X H**=**J**+ ¶**D**/ ¶ t (A current density**J**arises from a magnetic field)
ii)

**Ñ****X E**=**-**¶**B**/ ¶ t (A magnetic field can arise from an electric field)
iii)

**Ñ****·****B**=**0**(There are no magnetic monopoles)
iv)

**Ñ****·****D**= r**(Charges are conserved)**
In
addition, there are three “constitutive relations” that allow any of the above
vectors to be re-cast in slightly different forms:

v)

**D**= e**E**
vi)

**B**= m**H**
vii)

**J**= s**E**

In
the equations above,

**H**represents the*magnetic field intensity*,**B**is the*magnetic induction*,**E**the*electric field intensity*,**D**the*displacement current*, and**J**is the*current density*. The constants, e and m, denote the permittivity and the magnetic permeability – each for media. In vacuo, the constants used are: e_{0}and m_{0}and the speed of light can be expressed: c = 1/ Öe_{0}Ö m_{0}.
One
can also obtain the wave equations from the Maxwell differential (vector)
equations. For example, take:

**Ñ**

**X H**=

**J**+ ¶

**D**/ ¶ t

Take the
current free (J=0) case and we know:

**D**= ε

_{o }

**E and**

**B**= m

_{o}

**H**

Take the curl
of both sides of the vector equation in

**H**:**Ñ**

**X**

**Ñ**

**X H**=

**Ñ**

**X**(¶

**D**/ ¶ t)

And:

**Ñ**

**X**(¶

**D**/ ¶ t) =

**Ñ**

**X**(ε

_{o }¶

**E**/ ¶ t ) = ε

_{o}[

**Ñ**

**X**(¶

**E**/ ¶ t )]

But:

**Ñ**

**X**(¶

**E**/ ¶ t) = - ¶

^{2 }**B**/ ¶ t

^{2}*Where*

**:****- ¶**

^{2 }**B**/ ¶ t

^{2}**m**

*= -*_{o}¶

^{2 }**H**/ ¶ t

^{2}
Whence:

**Ñ**

**X**

**Ñ**

**X H**=

**ε**

_{o}(

**m**

*-*_{o}¶

^{2 }**H**/ ¶ t

**) = -m**

^{2}_{o}ε

_{o}(¶

^{2 }**H**/ ¶ t

**)**

^{2}
But
by a vector identity:

**Ñ**

**X**

**Ñ**

**X H**=

**Ñ**

**·**

**Ñ**

**·H -**

**Ñ**

^{2 }H
But:

**Ñ****·H =**1/ m_{o}**(****Ñ****·B**)**= 0****So: -**

**Ñ**-m

^{2 }H =_{o}ε

_{o}(¶

^{2 }**H**/ ¶ t

**)**

^{2}
Or:

**Ñ**m^{2 }H =_{o}ε_{o}(¶^{2 }**H**/ ¶ t**)**^{2}
Which
is one of the wave equations in terms of

**H**.
Writing
all the component wave equations out:

¶

^{2}H**/ ¶ x**_{x}**= m**^{2}_{o}ε_{o }¶^{2 }**H****/ ¶ t**_{x}^{2}
¶

^{2}H**/ ¶ x**_{y}**= m**^{2}_{o}ε_{o }¶^{2 }**H****/ ¶ t**_{y}^{2}
¶

^{2}H**/ ¶ x**_{z}**= m**^{2}_{o}ε_{o }¶^{2 }**H****/ ¶ t**_{z}^{2}
1.

**Solar Physics formulation of Electrodynamic Laws:**

In
solar applications,

**D**and**H**are seldom - if ever – used, and neither are m and e . For example, for Maxwell equation (i) one is more apt to make the following changes:
i’)

**Ñ****X**__(__**B**/m_{0}**)****=****J**+ ¶ (e_{0}**E**) / ¶ t
leading directly to:

**Ñ**

**X B**= m

_{0}[

**J**+ ¶ (e

_{0}

**E**) / ¶ t ]

If: ¶ (e

_{0}**E**) / ¶ t ® 0, Then:**Ñ**

**X B**= m

_{0}

**J**or:

**curl B**= m

_{0}

**J**

This is the solar version
of

*Ampere’s law*.
The solar version of
Ohm’s law is generally modified for finite conductivity, s to obtain:

**J**= s (

**E + v X B**)

However, there are
special cases for which one defines a “frozen in” condition. Obtaining the
particular condition depends on the laboratory, solar or space physics
situation and context.

Adopting the
first condition and allowing v = 0 we find:

**Ñ**

**X B**= m

_{0}[s

**E**]

Now, take the curl of
both sides:

**Ñ**x

**(**

**Ñ x B**) = m

_{0}s

**(**

**Ñ x E**)

Also, we have from a well
-known vector identity:

**Ñ**(

**Ñ**

**·**

**B**) -

**Ñ**

**·**

**Ñ**

**B**= m

_{0}s (

**-**¶

**B**/ ¶ t )

From Maxwell’s divergence
free equation (iii):

(

**Ñ****·****B**) = 0
So the previous equation (on substitution) reduces to:

¶

**B**/ ¶ t =**Ñ**^{ }^{2}**B /**m_{0}s
Which has the form of the
standard diffusion equation:

¶

**r / ¶ t = D****Ñ**^{ }^{2}**r**
Where D denotes the
diffusion coefficient, in this case:

D = 1/ m

_{0}s
This has dimensions of
(length)

^{ 2}divided by time. In effect, a sample length scale for consideration would have:
L’ = Ö (t/ m

_{0}s )
Such that for all times t’ < < t, the
magnetic field and the plasma can be considered to move as one. This defines
the term frozen in.

2.

**Solar Physics formulation of frozen in condition & Ampere’s Law**
In solar physics the
formulation is a bit more complicated and begins with re-casting Ohm’s law for
current density:

**E**=

**J /**s - (

**v X B**)

Again, take the curl of
both sides of the equation:

**Ñ**

**X E**=

**Ñ**

**X J**

**/**s

**-**

**Ñ**

**X**(

**v X B**)

But:

**Ñ****X B**= m_{0}**J**so that:**J**=

**Ñ**

**X B /**m

_{0}

Hence:

**Ñ**

**X E**=

**Ñ**

**X**

**Ñ**

**X B /**m

_{0 }s

**-**

**Ñ**

**X**(

**v X B**)

This leads to the end
result:

¶

**B**/ ¶ t =**Ñ**^{ }^{2}**B****/**m_{0 }s +**Ñ****X**(**v X B**)
Note that the first term
on the right is exactly analogous to the plasma diffusion term derived in the
lab plasma context. One can think of it in terms of resistive “leakage” or the
diffusion of magnetic intensity across the conducting fluid. The second term containing the velocity is a
convective term and one can think of it in the solar context as the appropriate
representation of the frozen in condition for magnetic field lines in solar
plasma.

Whichever term becomes
dominant will depend on the time and length scales and the magnitude of the
magnetic Reynolds number Â

_{m }. If Â_{m }>> 1 then diffusion can be ignored, otherwise it can’t. It is commonly expressed:
Â

_{m}= L V_{A}/ h
Where L is a typical
length scale for a given solar environment, V

_{A}is the Alfven velocity and h is the magnetic diffusivity. The infinitely conducting condition applies for h -> 0. This implies zero electrical resistance so if magnetic field lines aren’t cut, e.g. as shown here:**Plasma flow cutting field line**

Then the field lines must be frozen into the plasma. Hence, the
frozen-in condition, high magnetic Reynolds number and infinite conductivity
all mean the same condition for solar plasma.

*Ampere’s law in the solar context*:

A more
general form of Ampere’s relation is given by:

∮

**B**d ℓ = ò ò_{S }_{ }**J**dS = m_{0}**I**
Where I denotes the ‘enclosed’
current, ∮ denotes the closed line integral around the
closed curve C and ò ò

_{S }denotes the second surface integral over S enclosed by C.
In S.I. units it then becomes:

2pr

**÷**B**÷**= m_{0 }I
So that:

**÷**B**÷**= m_{0 }I/ 2pr
Where m

_{0 }= 4p x 10^{-7}H/M
The total current can also be expressed, in terms of the
z-component of magnetic intensity;

I

_{T}= 2 pr B_{z}/ 0.012
Ampere’s law in its vector identity form is often written:

**curl H**=

**J**or

**curl B =**m

_{0 }

**J**

where ‘

**curl**’ is a complex mathematical operation using partial derivatives for a certain coordinate system, such that for example:**curl**

**B**= [1/r

**e**

_{r}

**e**

_{f}

**e**

_{z}]

[¶/¶r
¶/¶f ¶/¶ z]

[ B

_{r}rB_{f}B_{z}]
For cylindrical coordinates, such as applicable to
solar loops to a fair approximation. This can also be further simplified for
one-dimensional modeling, cf.

with
B = (0, B

_{f}, B_{z })**provide**d: ¶/¶f = ¶/¶ z = 0.
Then:

**curl B**= [0 -¶ B_{z}/¶r_{ }1/ r ¶//¶r_{ (}rB_{f}_{)}]
which
is the exact truncation value of the Lundvist Bessel function solution.

In
solar physics situations, the force –free assumption dominates so that:

**J X B**= 0

In
general, we incorporate the “force –free” parameter (a) into the solar version of Ampere’s law such
that:

**curl B**= a

**B**

And since:

**curl B**=**m**_{0 }**J**
Then: a = m

_{0 }**J**/**B**and**J**= a**B****/**m_{0}
Thus:

_{}**Curl B**=

**Ñ**

**X B**= a

**B =**(m

_{0 }

**J**/

**B**)

**B**

This
is the mathematical starting point for the treatment of an evolving force-free
field. For a potential
(current-free) field a = 0 so that

**Ñ X B**= 0. This is the simplest case, i.e. in which no magnetic free energy is stored.
As an example, consider using the basic 1-D curl:

**curl B**= [0 -¶ B

_{z}/¶r

_{ }1/ r ¶/¶r

_{ (}rB

_{f}

_{)}]

From
the preceding curl elements:

-¶ B

_{z}/¶r_{ }= aB_{f}
1/r ¶/ ¶r (r B

_{f}) = aB_{z}
Whence,
for variation in one quantity:

B

_{f}**= 1/a (****d****B**_{z}/ dr )
and

1/r d/dr (r

**B**_{f}**) - a B**_{z}= 0
Now,
substitute the top equation into the bottom and multiply through by (-a) to get:

1/r d/dr (r d

**B**_{z}**/ dr ) + a**^{2}**B**_{z}= 0
which
is a form of Bessel’s differential equation. If
B

_{z}is finite on the r = 0 axis, then the solution may be written (Lundqvist, 1951)[1]
B

_{z}= B_{o}J_{o}(aR) and B_{f}= B_{o}J_{1}(aR)
where
J

_{o}(aR) is a Bessel function of zero order, and J_{1}(aR) is a Bessel function of first kind, order unity.
Note for the special case a
= 0 we get what is called the current-free condition for which there is no
residual free energy to be extracted. On
the other hand, for any region for which a
> 0, there exists MFE to be extracted, e.g. for solar flares.

Problems:

1)Take the electric field

**E**to be in the x-direction and write out an expression for**curl E**.
2)For E in three dimensions (x, y, z) show that:

**div curl E**= 0;

3)For a particular solar active region the magnetic diffusivity
is h »
327.6 m

^{2}/s
If
the length scale is L » 10

^{7}m and the Alfven speed is V_{A}= 10^{3}m /s, then find the magnetic Reynolds number for the region. From this assess whether the magnetic field is frozen in or not.
4)
The vector potential

**A**is often written as:**B**=**curl A**
Write out the full mathematical form for

**curl A**in rectangular coordinates.
5) (a) A solar loop has an estimated diameter of 1.1 x 10

^{9}cm. If the longitudinal magnetic field (estimated by vector magnetograph) is B_{z}» 0.1 T, estimate the total current.
(b)
A steady current
I flows through a hollow cylinder of radius a
and is uniformly distributed around the tube. Let r be the distance from
the axis of symmetry of the tube to a given point.

Find the magnitude of the magnetic field B at a point inside the tube. What is the magnitude of the magnetic field B, at a point

Find the magnitude of the magnetic field B at a point inside the tube. What is the magnitude of the magnetic field B, at a point

*outside the tube*?
6) For the problem 5(a), using the same quantities, estimate the
force free parameter, a. Typical solar values of a associated with coronal loops are of
magnitude » 10

^{-10 }m^{–1}.**How does the value you obtained compare?**^{ }
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