Plasma Physics I

Chapter 1: Mathematical preliminaries

Unavoidably, when developing theoretical physics, one needs to use particular suitable mathematical tools which allow to further develop the basic ideas. Generally, the principal tools are so-called tensors, entities with particular properties with which a student should be, more or less, already familiar. In particular, scalars numbers, vectors, and double tensors (represented by matrices) are known, and for now, we will introduce the notation we will adopt in the following notes:

• Scalar numbers: a, A, …
• Vectors: a, A, a_i, aⁱ, …
• Double tensors: a, A, a_ij, …

Those entities, as physical quantities, can, in general, vary depending on the value of other quantities; in particular, when the dependence is on the position in space, the quantity is called a field. If the dependence is also on time, the field is referred to as a dynamic field. We will see plenty of examples in this course.

A tensor is generally built up by different components, depending on the number of dimensions we consider; in physics, besides some very peculiar cases, we refer to three spatial dimensions and one additional time one; hence we will have up to 4 components for a vector and up to 16 for a double tensor.

These components have to be defined with respect to a frame of reference, that is the basis vector used to define any point in the space we consider. A lot of frames are known and used in physics, but the most common is the orthonormal cartesian one; such a frame has constant basis vectors, eⁱ, which are mutually orthogonal and have unitary length; in other words, they satisfy the following relation:

eⁱ ⋅ e^j =δ_ij

We will see that through this statement a lot of useful correlations can be found. But why tensors are of so much importance in physics? The question is not as philosophical as one may think, but it concerns the very same definition of a tensor. Moreover, it is based on the fact that, given a certain frame of reference, a relation between two tensors remains the same if the frame is changed. Physically speaking, any physical law written in tensor form is invariant with respect to the change in frame of reference; it is said that a physical law must be covariant.

Tensors in physics

Let us now consider a Euclidean tridimensional space, which we will call E₃, and tensors in it, focusing on the simplest ones (except for scalars), that are the vectors. A very important property of a vector x = (x₁, x₂, x₃) is the way in which one defines its length; in E₃ this is defined using Pythagoras's theorem:

|x| = √(x₁² + x₂² + x₃²) = √(Σ_ix_i²)

in which we both explicitly wrote the operation, then it has been written in index form -implying the summation- and, finally, in matrix form.

Defining O as a general transformation acting on a vector such that x' = Ox, it can be represented by a matrix O = [O_ij] and we can write the transformation of x in matrix notation: x'_i = O_ijx_j (in which the summation is intended over the double index, called dummy index). By a simple algebraic calculation, one can show that, in order to have that the length of x' is equal to the one of x, hence that it is an isometry, we must require that O satisfies the property of being orthogonal, that is O^TO = I.

This request corresponds to the one requiring that the determinant of O is equal to ±1, depending on the kind of orthogonality: proper orthogonality is det(O)=1; improper orthogonality if det(O)=-1.

A different transformation for the vector can be found if O is component dependent; taking the derivative of x'_i, we can write:

x'_i,^j = O_ijx,^j = (∑_jO_ijx_k),^j

Such transformation leads to a more general definition of a vector in E₃.

One might ask which is the behavior of a vector in E₃ space under some transformation. For example, under inversion of the coordinates we would expect that a vector x transforms into -x; however, such is not the case for every vector (think about a vector that can be written as the cross product of two vectors). Vectors that change sign under inversion are called polar vectors, those which don't are called axial vectors; in the following, we will see examples of such vectors, such as the electric field, a polar vector, and the magnetic field, an axial vector.

If a further dimension is added to a vector, we obtain a double tensor. This can be done through the so-called direct product, which applied to two vectors gives a tensor:

[a_ij] = a_ib_j

From this definition, one can show that a double tensor transforms as:

O^T T O = a'_ij = O_ikO_jla_kl, if a'_i = O_ika_k and b'_j = O_jlb_l

Let us now see some operations between tensors in order to introduce the notation we will use:

• Scalar product: a⋅b = δ_ija_ib_j
• Cross product: a×b = ϵ_ijka_jb_k
• Derivation operators: a⋅∼ = a∼,ⁱ, a⋅aⁱ, ⎯aⁱ, a⋅aⁱ, a⋅aⁱ, a⋅aⁱ
• And their combinations: ∇f, ∇×f, ∇⋅f××f

Other examples of differential operators are the so-called series expansions and, in particular, Taylor expansion is of great interest:

f(x_i) = f(0) + x_if,ⁱ + ½x_ix_jf,^ij + ...

For what concerns integral operators instead, we can have one, two, or three-dimensional integrals. Here we will simply recall some important theorems:

• Divergence theorem (Gauss theorem): consider a vector field A(x) sufficiently regular and a volume V enclosed in a surface S; we can write the following statement:

∫ (∇⋅A) d³x = ∫_S A⋅dσ

The right-hand side is what in physics is called flux; we will see that this theorem is very useful to define a divergence-less quantity.

• Curl theorem (Stokes theorem): consider a vector field A(x) and an open surface S enclosed in a line C; then

∫ _σ (∇×A)⋅dσ = ∫_C A⋅d l

The right-hand side is called circuitation of A; this theorem is useful to define a curl-less quantity.

General curvilinear coordinates

There may be various reasons for which the cartesian system of coordinates is of little use. The first thing is that not in all systems the rule that the basis vectors are orthonormal is true; secondly, the fact that such a rule can be local and not global, meaning that it holds for a point and not at all.

A first example of a different set of basis vectors is the well-known system of polar coordinates, which introduces a dependence on the angle to the vectors, making them coordinate-dependent; a further example is the system of oblique coordinates, very useful to introduce the concept of covariant and contravariant component of a vector. In fact, one can write the vector a as:

a = a¹e₁ + a²e₂

with components a¹, a²; the other way around, one can take the components obtained by projecting orthogonally the vector on the basis:

a¹ = a⋅e¹, a² = a⋅e²

The first kind is called contravariant components, while the second covariant; they are, in general, different - equal in orthonormal cartesian coordinates.

Typically, given a set of basis vectors, one can generalize the relation of orthonormality, written for the cartesian orthonormal frame, by introducing the so-called metric tensor, g_ij, so that:

eⁱ⋅e^j = g_ij

such tensor can be coordinate-dependent and gives information about the metric of a given part in space, that is the basis vectors relation.

Consider now a vector A = Aⁱe_i, and the fact that Aⁱ can depend on the coordinates; then the derivative of Aⁱ with respect to the coordinate qⁱ will be composed of two terms:

∂Aⁱ/∂qⁱ = e_i∂Aⁱ/∂qⁱ + Aⁱ∂e_i/∂qⁱ

Using the properties of the metric tensor, however, we can compute the second term; in fact:

∂e_i/∂qⁱ = Γⁱ_jke^k

where:

Γⁱ_jk = (g^ij,_k + g^ik,_j - g_jk,ⁱ)/2

contains the derivatives of the metric tensor and the term g^kl, defined such that g^klg_lj = δ_i^j - namely the inverse of the metric tensor. Γⁱ_jk is called connection symbol or Christoffel symbol; such mathematical structures are needed in general relativity and curved spaces.

Chapter 2: Recalls of electromagnetism

Given the known concepts that lead to Maxwell equations, which are the pillars of electromagnetic theory, we will try to write them in a more general form, without considering a particular frame of reference. To do that we will assume to work in vacuum -for now-, meaning that all the charges and current can be explicitly written. Said that, let's start from statics.

Electrostatics and magnetostatics

Electrostatics

During the second half of the eighteenth century, Charles Augustin de Coulomb noticed that a force is mutually experienced by two electrically charged bodies; such action is proportional to the product of the charges and inversely proportional to the square of their relative distance. This force can be represented as a vector, directed along the joining of the bodies and it can be written as:

F = k (q₁q₂/r²) u_r

in which k is an arbitrary constant we did not define yet. The following step is to explain why such force at a distance is felt by both the charges. One can define a so-called electric field as:

E = F/q₁ = k (q₂/r²) u_r

The third natural step would be to consider more than one charge producing the electric field; in this case, with an arbitrary number N of point charges q_i, one can write:

E(x) = k Σ_i=1^N (q_i/(|x - x_i|³)) (x - x_i)

If the charges are so small and so numerous that they can be described by a charge density ρ(x'), the sum is replaced by an integral:

E(x) = k ∫ (ρ(x')(x - x')/|x - x'|³) d³x'

from which the summation can be obtaining writing the density of charges as a sum of weighted Dirac delta functions: ρ(x) = Σ_iq_iδ(x - x_i(t)) - we introduced a time dependency since, in principle, a charge is able to move in space.

Notice that the vector factor in the integrand, seen as a function of x, is a negative gradient:

∂(ρ(x')(x - x')/|x - x'|³) = -∇(x - x')/|x - x'|

Introducing it in the definition of the electric field we can take the gradient outside the integral and write:

E(x) = -k ∇∫ (ρ(x')/(|x - x'|)) d³x'

Here the electric field (a vector) is written as the gradient of a scalar. Since one function of position is easier to deal with than three, it is worth concentrating on the scalar function itself; thus, we define the scalar potential as:

Φ(x) = k ∫ (ρ(x')/(|x - x'|)) d³x'

so that:

E = -∇Φ

Notice that from this expression results that the curl of E is zero, since the curl of a gradient is always zero: ∇×E = 0. Furthermore, from Stokes' theorem, we have that the circuitation of E is null.

If now we consider a closed surface and a point charge in it, the flux of the electric field would be:

∫ E⋅dσ = k (q₁/r)³ = 4π

Such correlation, properly modified with the divergence theorem and expression the charge in terms of density, is called Gauss' theorem and it is the first Maxwell's equation we encounter:

∇⋅E = 4πρ

Magnetostatics

The simplest situation in which electric currents are flowing is the one with two infinitely long parallel wires; here we define the current, I, as the derivative in time of the charge flowing in the wire. Similarly to charges, one can define a current density, J, such that:

I = dq/dt = ∫ J⋅dσ

In such situation, one observes a force between two infinitesimal pieces of wire, which is directly proportional to the product of the currents and inversely proportional to the distance between the wires:

dF/dl = (2k₂/r²) I I'

in which we set the proportionality constant equal to 2k₂, for reasons we will see later.

The same as we did introducing the electric field can be done in this case, dividing the force by I' one can indeed find what is called the magnetic field, of intensity:

B = (2k₂α/r)

where α is another proportionality constant -we could have introduced something similar when we derived the electric field, however, we exploited this degree of freedom to set the constant equal to one. Nothing we said about the direction of either the force and the field, yet; the latter can be written as:

dB = (2k₂Iα/r³) dl×r

while the force is:

dF/dl = (I/α₂r³) dl×B

It can be useful to write these results in terms of the current density; noting that Idl = Jd³x we can write:

B(x) = (k α₂/r³)∫ J×d³x'

which is in some way very similar to the expression for the electric field we found before. Using again the fact that the vector factor in the integrand is a gradient, and using the vector identity:

∇×(a×b) = a(∇⋅b) - b(∇⋅a)

we obtain:

B(x) = k α₂∇×A

in which we defined the vectorial quantity:

A = ∫ (J/(|x - x'|)) d³x'

(again very similar to the one for E). Since now the magnetic field is equal to a curl, we get what it is the second Maxwell's equation in our path:

∇⋅B = 0

We may derive the curl of B, assuming that the divergence of the current density is null; this expression:

∇×B = 4πα₂J

called Ampere's law, holds just in magnetostatics, since we are also assuming that the charges don't move, which is not true in dynamics.

Electromagnetic dynamics

Now, evidently, electric and magnetic phenomena are related: charges produce electric forces, which move other charges and produce currents, which generate magnetic fields and so on. This link was seen by Maxwell, who was able to generalize the expression for the curl of B adding a term; this can be found using the fact that the charge density derivative in time can be written as:

∂ρ/∂t = ∂⋅E / 4k₁π

and the so-called continuity equation for the electric charge in vacuum,

∂ρ/∂t + ∂⋅J = 0

Using these results, we obtain the third Maxwell equation:

∇×B = 4πα₂J + ∂α₂k₁E/∂t

such correlation has little importance in statics, but is of major importance in dynamics.

The fourth and last Maxwell's equation has to be found from the work of Faraday. He observed that a magnetic field which changes in time, crossing a surface, is able to produce an electric field; in fact, it can be shown that the electromotive potential, ℰ, produced on the contour of the surface, is proportional to the derivative in time of the magnetic flux, that is:

ℰ = -k₃∂Φ_B/∂t

Combined with the definition of circuitation and flux, through Stokes' theorem, one can obtain the fourth Maxwell's equation:

∇×E = -k₃∂B/∂t

Constants and unit system

One has certainly noticed that we left almost all the degrees of freedom arising from the selection of the proportionality constant untouched. Our next step is to derive some correlations between the constants introduced, starting from considering the ratio between the electric force F and the infinitesimal magnetic force, dF/dl; thus:

F₁/df₂/dl = k₁q₁q₂/r² / k₂I I'/(r²)