Electromagnetism: Maxwell Equations, Wave Propagation and Emission

Chapter 1

Prologue

Most physical phenomena are fundamentally electromagnetic. This makes electromagnetism a basic theory in many branches of physics (solid state physics, electronics, atomic and molecular physics, relativity, atmospheric physics, etc.) also in some other sciences and most technologies.

Although physics is an experimental science, it uses mathematical language to formulate its theories and its laws and analyze their consequences. Electromagnetism is a typical theory that is impossible to formulate without extensive use of vector analysis, differential equations, complex analysis, etc. The use of mathematics can even lead to the prediction of new physical laws and new phenomena (the discovery of the electromagnetic waves by Maxwell is a typical example). However, only experiments can decide whether a particular solution or prediction and even the whole theory is acceptable. Until now, no experiment has contradicted electromagnetic theory, both on the macroscopic scale and the microscopic scale (nuclear, atomic or molecular).

Although permanent magnets and electrification by rubbing were known in antiquity, scientific observations of magnetism began around 1270 with the French army engineer Pierre de Marincourt. The observation of electric effects began much later with the French botanist C. Dufay around 1734. Contrary to the gravitational interaction between masses, the large majority of objects around us are globally neutral and, if they become charged, they discharge rapidly in the surrounding air. The scientific study of electricity started with Franklin (1706–1790), Priestley (1733–1804), Cavendish (1731–1810), Coulomb (1736–1806), Laplace (1749–1827), Ampère (1775–1836), Gauss (1777–1855), and Poisson (1781–1840) who formulated the laws of electricity and magnetism. Faraday (1791–1867) introduced the notions of influence and fields and discovered electric induction, which allowed the large-scale production of electricity. Electricity and magnetism were unified in a single theory by Maxwell in 1864. This long itinerary led to the present technological era with the considerable influence of electromagnetism and its consequences on our industrial, economical and cultural environment.

In this chapter, we introduce some basic mathematical methods and some general invariances and symmetries that we use in the formulation of any theory and especially electromagnetic theory.

1.1. Scalars and vectors

The basic elementary concepts in the formulation of physical theories are position and time. The position is specified by the coordinates with respect to a reference frame Oxyz, supported by a material body and represented by an origin O and three mutually orthogonal axes. Although these concepts seem to be simple, their analysis poses deep practical and philosophical questions even in classical mechanics. In modern physics, their analysis has been one of the corner-stones of the special theory of relativity (see Chapter 13), general relativity, and quantum theory.

Some physical quantities are determined by a single algebraic quantity with no characteristic orientation. Mass, time, temperature, and electric charge are examples of such quantities; these are scalar quantities. They may be strictly positive (mass, pressure, etc.), positive or negative (position along an axis, potential energy, electric charge, etc.), or even complex (wave function, impedance, etc.). Other physical quantities A are specified, each one by a positive magnitude A and an orientation; these are said to be vector quantities. Displacement, velocity, acceleration, force, electric field, magnetic field, etc., are examples of vector quantities. A more precise definition of a vector quantity is given in section 1.2.

A vector A is conveniently specified by its Cartesian components Ax, Ay and Az with respect to a frame Oxyz (Figure 1.1a). We may write ch1_eqa02_01 , where ex, ey and ez are the unit vectors of the axes Ox, Oy and Oz; they are the basis vectors of the reference frame Oxyz. To simplify the writing of summations, we use the numbers 1, 2 and 3 instead of x, y and z to label the components and we write

[1.1] ch1_eqa02_02

The component A1, for instance, is the projection of A on the axis Ox. It is well known that the decomposition [1.1] is unique.

The product kA of a scalar k and a vector A is the vector kA parallel to A and of magnitude k times the magnitude of A. The components of kA are simply those of A multiplied by k. The resultant (or sum) (A + B) of two vectors A and B is defined by the usual parallelogram rule (Figure 1.1b). The components of (A + B) is simply the sum of the corresponding components of A and B:

[1.2] ch1_eqa03_01

Figure 1.1. a) Cartesian components of a vector. b) Sum of two vectors A and B. c) The cross product A×B. d) The triple scalar product (A×B).C

Scalar product

The scalar product (or dot product) of two vectors A and B, written as A.B, is the product of their magnitudes and the cosine of their angle θ. Thus, the scalar product of a vector A by itself, written as A², is the square of its magnitude, A² = A². We note that the scalar product is linear in A and B. In the case of the basis vectors, we have eα ² = 1 and eα.eβ = 0 if α≠ β. Using the Kronecker symbols δαβ, we may write:

[1.3]

This allows us to write the scalar product of A and B in terms of their components:

[1.4]

The unitary vector eB in the direction of a vector B is obtained by dividing B by its magnitude

[1.5] ch1_eqa03_04

If a vector A forms an angle θ with B, the projection of A on B is

[1.6] ch1_eqa04_01

A may be written as ch1_eqa04_02 , where ch1_eqa04_03 is parallel to B and ch1_eqa04_04 is normal to B:

[1.7]

Cross product

The cross product (or vector product), designated by A × B, is the vector

[1.8] ch1_eqa04_06

where n is the unit vector that is normal to the plane containing the vectors A and B and oriented according to the right-hand rule: if the thumb and the forefinger are in the directions of A and B, respectively, the middle finger points in the direction of A × B (Figure 1.1c). Note that the area of the parallelogram of sides A and B is just the magnitude of A × B.

Contrary to the scalar product, the cross product A × B is not commutative: it is odd in the exchange of the vectors: (A × B) = − (B × A). The cross product of two parallel (or antiparallel) vectors is equal to zero because θ = 0 (or θ = π). It may be verified that eα × eβ = eγ, where (α, β, γ) is a circular permutation of (1, 2, 3), that is

[1.9] ch1_eqa04_07

This allows us to write the components (A×B)α = Aβ Bγ − Aγ Bβ, that is,

[1.10]

We may also write the cross product as a determinant

[1.11] ch1_eqa04_09

Triple scalar product

The so-called triple scalar product of three vectors is defined by U = (A × B).C. It is invariant in a circular permutation of the vectors and odd in the exchange of any two vectors. It can be interpreted as the volume of the parallelepiped of sides A, B, and C with a positive sign if the trihedron A, B, C, taken in this order, is righthanded and a negative sign otherwise (Figure 1.1d). It may be expressed as the determinant of the components

[1.12] ch1_eqa05_01

Differentiation of vectors

The differentiation rules for sums and products of functions hold for vectors. To simplify the notation, the derivatives ch1_eqa05_02 etc., are written as ch1_eqa05_03 etc.If a vector A depends on time, the components Aα depend on time. Thus, if the basis eα is time-independent, the differential of A and its derivative with respect to time are

[1.13]

If the basis vectors depend on time, we must write

[1.14] ch1_eqa05_05

1.2. Effect of rotations on scalars and vectors

The choice of the origin O and the orientation of the axes of reference are completely arbitrary and observers in different places and different times often use different reference frames, different origins of time and even moving frames, relative to each other. Although these observers may find different coordinates and different time for any given event, it is evident that they must find the same laws for any physical phenomenon (otherwise, physics would not be a science at all). This is known as the relativity principle. Thus, it is necessary to know how physical quantities are related in different frames Oxyz and ch1_eqa05_06 . A physical quantity that depends on position ch1_eqa05_07 and time t in Oxyz is a field, which we write as f(r, t) or f(xα, t), where xα is a shorthand notation for the coordinates x, y and z of r.

We consider two parallel frames Oxyz and ch1_eqa05_08 , such that the origin ch1_eqa05_09 has a fixed position ch1_eqa05_10 (of coordinates xo, αwith respect to Oxyz), the position of an event ch1_eqa05_11 with respect to ch1_eqa05_12 is related to its position r with respect to Oxyz by the equation

[1.15]

This is a simple translation in space. Any field, whatever its nature, must be specified by equal values f(r, t) and ch1_eqa05_14 in these frames, thus

[1.16] ch1_eqa06_01

We consider now the more interesting case of reference frames related by rotations. The basis vectors e’ β of the frame ch1_eqa06_02 are related to the basis eα of the frame Oxyz by a linear transformation

[1.17] ch1_eqa06_03

where R is a 3×3 matrix and ch1_eqa06_04 is its inverse. Writing A = Σα Aα eα and expressing the eα in terms of the e’ β by using [1.17], we find

[1.18] ch1_eqa06_05

Comparing with ch1_eqa06_06 we deduce that

[1.19] ch1_eqa06_07

In particular, these transformations hold for the coordinates that are the components of the vector r. Using vector notation, we write

[1.20] ch1_eqa06_08

The transformation R conserves the scalar products (and in particular the magnitude of vectors) if it is orthogonal (that is, its transposed ch1_eqa06_09 is equal to its inverse). In other words, it verifies the condition

[1.21] ch1_eqa06_10

where I is the unit matrix (that is, it has ch1_eqa06_11 as matrix elements).

A physical quantity f is a scalar if it is invariant in any rotation R. If it is a scalar field, it must verify the condition

[1.22] ch1_eqa06_12

This is the case of r² or any scalar function of r² ch1_eqa06_13

The three quantities Aα are the components of a vector A if they transform according to [1.19], exactly as the coordinates xα in any rotation R. The functions Aα(r) are the components of a vector field A(r), if they transform according to

[1.23] ch1_eqa06_14

1.3. Integrals involving vectors

Circulation of a vector field

The circulation of a vector field E in a displacement dr = dx ex + dy ey + dz ez is E.dr (Figure 1.2a). The work of a force is a typical example of circulation. The circulation of E along a curve ch1_eqa06_01 going from r to ro is the line integral

[1.24] ch1_eqa07_02

where dr is the infinitesimal displacement along the path ch1_eqa07_03 and ch1_eqa07_04 is the tangential component of E. The circulation is a scalar quantity defined as the limit of the sum of the scalar products ch1_eqa07_05 of the infinitesimal elements ch1_eqa07_06 and the fields En at these elements. To calculate the integral in the general case, a parametric representation of the curve x = x(u), y = y(u) and z = z(u) may be used, where u is any parameter with u and uo corresponding to the extreme positions r and ro. The components Eα become functions of u and the circulation becomes an integral over u

[1.25]

If the field has a uniform tangential component ch1_eqa07_08 along the path ch1_eqa07_09 , its circulation is ch1_eqa07_10 , where ch1_eqa07_11 is the length of the path. On the other hand, if E = E ez is uniform in the direction Oz, its circulation is ch1_eqa06_12

Figure 1.2. a) Circulation of E along a path ch1_eqa06_13 going from r to ro. If E is conservative, this circulation is equal to V(r) – V(ro) for any ch1_eqa06_14 b) Setting V(∞) = 0, V(r) is the circulation of E along an arbitrary path going from r to infinity. c) The flux of E through an infinitesimal surface dS. d) The flux through an open surface S bounded by an oriented contour ch1_eqa06_15

Flux of a vector field

Consider the integral over a surface S

[1.26] ch1_eqa08_01

where E(r) is a vector field, n is the unit vector normal to the surface S at the running point r, θ is the angle of E with n and ch1_eqa08_02 is the component of E in the direction of n. This integral is the flux of E through S. The flux ch1_eqa08_03 through the infinitesimal area ds is a scalar quantity and so is the flux. Note that we may write ch1_eqa08_04 where ch1_eqa08_05 is the normal component of E, or ch1_eqa08_06 where ch1_eqa08_07 is the projection of dS on the normal plane to E (Figure 1.2c). dΦ is positive or negative, depending on whether θ is acute or obtuse, and itvanishes if E is tangent to S. Note also that n has two possible orientations; by changing the direction of n, we change the sign of Φ. In the case of an open surface, which is bounded by an oriented closed curve ch1_eqa08_08 we choose n according to the righthand rule (Figure 1.2d). In the case of a closed surface S, we choose n oriented outward; Φ is then the outgoing flux.

The flux is additive both for the vector field and for the area. In the particular case of a field having a uniform component in the direction of n, its flux is ch1_eqa08_09 Another physically interesting case is that of a radial field ch1_eqa08_10 of a charge q. Its flux through a closed surface S is ch1_eqa08_11 where Ω is the solid angle of the cone, whose apex is at q and which is subtended by S: it is equal to 4π if q is inside S and equal to 0 if q is outside S .

1.4. Gradient and curl, conservative field and scalar potential

The work of a force F acting on a particle of mass m in a displacement dr is dW = F.dr. This work is transformed into kinetic energy if no other force acts on the particle. Conversely, to displace the particle without acquiring kinetic energy dUK, an external agent must exert a force ch1_eqa08_13 and supply a work ch1_eqa08_14 If the force is conservative, this work is transformed into potential energy dUP of the particle in the field of force F. This analysis can be repeated for any vector field E. Its circulation along a path ch1_eqa08_15 going from r to ro depends in general on r and ro and also on the path ch1_eqa08_16 Its circulation on a closed path is not necessarily equal to zero. The differential form dx E1 + dy E2 + dz E3 is a total differential if the components Eα are the partial derivatives of a scalar function –V where is called the scalar potential corresponding to the field of force F. Then, we have ch1_eqa08_17 and ch1_eqa08_18 which we write in the vector form

[1.27] ch1_eqa09_01

The vector differential operator ∇ is called nabla or and ∇V is the gradient of V. It may be shown that the gradient of any scalar function V is a vector. In this case the circulation [1.24] becomes

[1.28] ch1_eqa09_02

In this special case, the circulation between two points is equal to the drop of the potential. It depends only on the points r and ro for any path C connecting these points (Figure 1.2a). In the case of a closed path ch1_eqa09_03 the circulation vanishes.We say that the field E is conservative. For instance, in the case of a uniform field E, the potential is ch1_eqa09_04 and in the case ch1_eqa09_05 where Vo is an arbitrary constant. In the last case it is convenient to assume that V vanishes at infinity, hence Vo = 0, and we may interpret V(r) as the circulation of E along anarbitrary path going from r to infinity (Figure 1.2b). In the case where E is a conservative field of force F, we may write ch1_eqa09_06 where Up is the potential energy. The work of F along a path ch1_eqa09_07 going from r to ro is ch1_eqa09_08 and the work of F along a closed path vanishes.

To know whether a vector field E is conservative, we do not have to evaluate the circulation on all imaginable paths. We may use the important property that the partial derivatives of a function are independent of the order of differentiation. If E is conservative (that is, ch1_eqa09_09 ), the equation ch1_eqa09_10 may be written as ch1_eqa09_11 Using the differential vector operator ∇, we define the vector

[1.29]

A vector field E is conservative if its curl is identically equal to 0, and it may be shown that the converse is true: if ch1_eqa09_13 E is conservative. In this case, we may define a potential V at each point r (see section A.7 in Appendix A)

Even if a vector field A is non-conservative, Stokes‘ theorem (see section A.8 of Appendix A) allows the expression of the circulation of A along a closed path ch1_eqa09_14 as the flux of ch1_eqa09_15 through any surface S bounded by ch1_eqa09_16

[1.30] ch1_eqa09_17

Note that the normal n is oriented according to the right-hand rule. We see from this theorem that, in the special case of a conservative field ch1_eqa10_01 its circulation along any closed path ch1_eqa10_02 vanishes and this is the definition of a conservative field.

1.5. Divergence, conservative flux, and vector potential

In general, the flux of a vector field B through the surfaces S bounded by a given closed contour ch1_eqa10_03 depends on the special choice of S and the flux through a closed surface is not necessarily equal to zero. We define the divergence of B as

[1.31] ch1_eqa10_04

It may be shown (see section A.7 of Appendix A) that, if ch1_eqa10_05 its divergence vanishes ch1_eqa10_06 and conversely, if ch1_eqa10_07 we may write

[1.32] ch1_eqa10_08

A is the vector potential. In fact, there are an infinite number of vectors A that correspond to the same B. They differ by a gradient term

[1.33] ch1_eqa10_09

because ch1_eqa10_10 The relation [1.33] is called gauge transformation.

Gauss-Ostrogradsky's theorem (see section A.9 of Appendix A) allows the expression of the flux of any vector field B through a closed surface S as the integral of ∇.B over the volume ch1_eqa10_11 enclosed by S

[1.34] ch1_eqa10_12

Note that to apply this theorem, the unit normal vector n must point outward S. We deduce that, if ch1_eqa10_13 the flux of B through any closed surface S vanishes. We say that B has a conservative flux. On the other hand, if ch1_eqa10_14 0 at a point M, the flux of B outgoing from any surface surrounding M is positive; thus, the field is divergent from M. On the contrary, if ch1_eqa10_15 this flux is negative and B is convergent at M.

1.6. Other properties of the vector differential operator

Here are some useful properties of the operator ∇ acting on scalar fields f(r) and g(r) and on vector fields A(r) and B(r):

[1.35] ch1_eqa11_01

[1.36] ch1_eqa11_02

[1.37]

[1.38] ch1_eqa11_04

[1.39] ch1_eqa11_05

[1.40] ch1_eqa11_06

[1.41]

[1.42] ch1_eqa11_08

where ch1_eqa11_09 is a scalar operator.

The successive application of ∇ on scalar and vector fields is very useful in physics. In Cartesian coordinates, if we evaluate the divergence of the gradient of a scalar function, we find

[1.43]

where the operator Δ, called Laplacian, is defined by

[1.44] ch1_eqa11_11

As ∇ is a vector operator, the Laplacian is a scalar operator. Acting on a scalar field, it gives a scalar field and, acting on a vector field, it gives the vector field

[1.45] ch1_eqa11_12

Thus, in Cartesian coordinates (and only in these coordinates), the components of ΔA are simply ΔAα. Other useful relationships may be obtained by successive applications of ∇:

[1.46] ch1_eqa11_13

[1.47] ch1_eqa11_14

[1.48] ch1_eqa11_15

1.7. Invariance and physical laws

By transformation, we mean a change of the coordinates or the variables of a system. A transformation is said to be continuous if it depends on parameters taking continuous values (as in the case of translations and rotations), otherwise it is said to be discrete (as in the case of reflections). A physical system is invariant in a transformation if it remains unchanged in the transformation (for instance, an infinite homogeneous medium is invariant in translations and a cone is invariant in rotations about its axis). A physical theory is invariant if it remains valid in the transformation (for instance, classical mechanics is invariant in the translation of time) and a physical quantity is invariant if it is unchanged in the transformation. An equation is said to be covariant in a transformation if it remains valid in the transformation (although the value of its terms may change in the transformation).

The invariance of a physical theory imposes some restrictions on the mathematical formulation of the laws and the allowed processes. A general principle, which was formulated by Noether, associates a conserved physical quantity with each invariance in a continuous transformation.

a) Invariance in geometrical transformations

Geometrical transformations are those of spatial coordinates and time, which conserve distances and intervals of time in classical physics. They include translations, rotations, and reflections. In a transformation (r, t) → (r′, t′), a physical quantity (or a field) f(r, t) becomes f′(r′, t′).

It is evident that physical laws do not depend on the origin of coordinates. In other words, an isolated system evolves in the same way, whatever its position in space (we say that the space is homogeneous). Mathematically, any physical law should not be modified if the positions r(k) of all the particles (k) of the system are modified by the same translation r′ (k) = r(k) + a. For instance, the interaction energy U12 of two particles located at r1 and r2 is invariant in the translations if U12 depends on R = r2 − r1and not on r1 and r2 separately. Thus, we must have U12 = U(R). Consequently, the forces that act on the particles are F1→2 = −∇2UE = −∇RU and F2→1 = −∇1UE = ∇RU, where ∇R means the vector differential operator with respect to the components of R. Thus, the invariance in translations implies that F12 = −F21 and, consequently, the conservation of the total momentum of a system of interacting particles such as electrically charged particles.

On the other hand, the physical laws do not depend on the orientation of the axes of coordinates. In other words, the space is isotropic. Mathematically, any physical law should not be modified if the reference frame is rotated. This requires, for instance, that the interaction energy of two particles is U12 = U(R), i.e. a function of the magnitude of R and not its direction. Consequently, the force F1→2 may be written as F1→2 = −∇RU = −(∂U/∂R) R/R. Thus, it is oriented along the line that joins the two particles. This implies that the total angular momentum of an isolated system of particles L = Σi mi ri × vi is conserved.

Physical laws obey another important invariance law: they do not depend on the choice of the origin of time. In other words, if an experiment is repeated in time, the result should be the same. Mathematically, any physical law should not be modified in translations of time t′ = t + to. This invariance requires that the potential energy of two bodies does not depend explicitly on time. Thus, the total energy of an isolated system is conserved if there are no dissipative forces.

To implement these invariance laws, the physical quantities must have well-defined transformation laws: they must be scalars, vectors, or other types of mathematical objects. A typical vector is the position r. Quantities, such as the force F and the electric field E(x, y, z), that transform exactly like r in rotations are vectors. For instance, in a rotation through an angle ϕ about Oz, they transform according to:

Here, the components of the field E are functions of the coordinates, E′ x(x′, y′, z′) and Ex(x, y, z) ≡ Ex(x′ cos ϕ− y′ sin ϕ, x′ sin ϕ + y′ cos ϕ, z′). A scalar is a quantity that is invariant in rotations, such as the distance r², the scalar product of two vectors

b) Invariance in reflections

To formulate physical laws, only right-handed reference frames Oxyz are usually used. However, nothing forbids to use systematically left-handed frames Ox′y′z′. Typical transformations of a right-handed frame to a left-handed one are reflections such as the total reflection r′ = −r (i.e. x′ = −x, y′ = −y and z′ = −z) and the reflection with respect to the Oxy plane (i.e. x′ = x, y′ = y and z′ = −z as in a mirror). The invariance of physical laws in reflections (that are discrete transformations) is not as evident as in translations and rotations (that are continuous transformations). However, the experiment shows that this invariance holds in mechanics, in electromagnetism and in the case of strong (nuclear) interactions. It is violated in the case of weak interactions (see section 1.9d).

Some vectors have components that transform in reflections exactly like the coordinates xα; these are said to be true vectors. Similarly, some scalars do not change in reflections; these are true scalars. This is the case for the distance ch1_eqa13_02 and the scalar product of two true vectors A.B. On the other hand, the cross product of two true vectors U = A × B transforms like r in rotations but, in reflections, the components Uα transform like xα with an additional change of sign. For instance, in the total reflection (xα→−xα), the components Uα remain unchanged (Uα→ Uα) and, in the reflection with respect to the Oxy plane (x → x, y → y and z →−z), the Uα transform according to the relations Ux →−Ux, Uy →−Uy and Uz → Uz. We say that U is a pseudo-vector. This is also the case for the cross product of two pseudo-vectors, while the cross product of a true vector and a pseudo-vector is a true vector. The scalar product of a true vector and a pseudo-vector is a pseudo-scalar: it is invariant in rotations but it changes sign in reflections; this is the case of the triple scalar product of three true vectors A.(B×C).

A physical law, written as a mathematical relationship between physical quantities, can be valid only if it is covariant in the preceding transformations. Thus, we may add, subtract or write equalities of quantities of the same type. It is not valid to add a vector to a pseudo-vector or write the equality of one component of two vectors without having the other components equal. For instance, the fundamental law of mechanics F = m d²r/dt² requires that F be a true vector (like r) and the definition of the potential energy by the relation dUP = − F.dr requires that Up and the energy in general be true scalars.

1.8. Electric charges in nature

Although matter is neutral on the macroscopic scale, it is comprised of charged and neutral particles. The experiment shows that, on the microscopic scale, the electric charge takes only discrete values (0, ±e, ±2e, ± 3e, etc.) that are integer multiples of the elementary charge

[1.49] ch1_eqa13_03

This quantization was established for the first time in 1913 by Millikan’s oil drop experiment (see Problem 14.3). The stable particles, which are the building blocks of matter, are the proton of charge +e, the electron of charge −e, and the neutron (which is neutral as its name indicates). The electrification by rubbing is simply a transfer of electrons from a body of low electronic affinity to another of higher affinity.

The equality of the charge of the proton and the charge of the electron in absolute values, i.e. the neutrality of the hydrogen atom, is verified by the absence of any deviation of this atom by electric or magnetic fields with a precision of 1 to 10²⁰. On the other hand, the electric charge of particles does not depend on their velocity or on physical conditions, such as temperature, pressure, etc., even in extreme conditions, as in the core of stars or in the early stage of the formation of the Universe. The electron and the proton are absolutely stable. It is not possible to eliminate one of them individually but an electron and a proton may interact and produce a neutron and a neutrino. Conversely, a neutron may decay into a proton, an electron and an antineutrino. More generally, physical, chemical or biological transformations may occur in an isolated system leading to the exchange of charged particles between the constituents of the system, the creation or the annihilation of pairs of oppositely charged particles, but the total charge of the system is conserved.

The quantization of electric charge and its numerical value as well as the equality of proton and electron charges in absolute values are not understood even today. On the macroscopic scale, the elementary charge is extremely small and often has no observable effect. For instance, a negative charge of 1 μC corresponds to 6 × 10 ¹² electrons and a current of 1 A carries 3.2 × 10¹⁸ electrons per second! When speaking of a point charge, it may be an elementary particle or a macroscopic object of small size compared to the dimensions of the system. It is often a very good approximation to consider an extended macroscopic charge as a continuous charge distribution.

a) Macroscopic bodies and molecules

Molecules and atoms are constituted by charged or neutral particles (electrons, protons, and neutrons). Their electric interactions are responsible for the cohesion of matter and most of its physical and chemical properties. Materials may be classified as conductors if some of the electrons are more or less free to move, insulators if the electrons are strongly bound to the atoms, and semiconductors whose conduction is intermediary between conductors and insulators. In solids and liquids, the spacing between atoms is of the order of the atoms‘ diameter (i.e. a fraction of a nanometer = 10−9 m). In gases, the molecules are separated by much longer distances. They are normally neutral at normal and low temperatures but some may become ionized by collisions, which become more and more frequent and energetic at high temperature. A gas may also become ionized if an energetic particle or radiation passes through it. A gas that is totally or partially ionized is a plasma.

b) Atoms, electrons, protons, and neutrons

The late 19th Century experiments have shown that atoms contain negatively charged electrons. To be globally neutral, the atoms must also contain positively charged particles, protons. To explain the stability of atoms, Thomson assumed that positive charges as well as the negative charges are distributed within a sphere of radius of the order of 10−10 m. However, Rutherford's experiment in 1911 showed that the positive charge is concentrated in a nucleus with a radius of the order of 10−15 m (see section 14.3). To explain the stability of the atom, in 1913 Bohr proposed a model in which the electrons maintain circular or elliptical orbits with a radius of the order of 10−10 m around the nucleus, bound by electric force. This orbital motion is similar to that of the planets around the Sun via gravitational force. Later, quantum theory abandoned this simple model in favor of a negatively charged electronic cloud around a positively charged nucleus. The state of the electrons in the atom is governed by the laws of quantum mechanics and, in principle, the properties of macroscopic matter can be deduced, but this is a difficult procedure.

The number of charged particles in matter is enormous. The number of molecules or atoms in a mole ¹ of substance is the Avogadro number NA ≈ 6 × 10²³ mol−1 and each atom of a given chemical element contains Z electrons and Z protons. The hydrogen atom (Z = 1), for instance, is formed by one proton and one electron. The helium atom (Z = 2) contains two electrons and two protons. If it contains no other particles, its atomic mass would be approximately twice that of hydrogen; experimentally it is four-times heavier. Thus the atomic nuclei must contain neutral particles. These particles, called neutrons, were observed by Chadwick in 1938 with a mass that is slightly higher than the mass of protons (i.e. about 1840 times the mass of the electron). The helium nucleus, called also alpha particle, is formed by two protons and two neutrons. The protons and neutrons, which constitute the atomic nuclei are referred to as nucleons. The atom is thus formed by Z electrons, Z protons, and N neutrons. Its mass is M ≅ Zme + Zmp + Nmn ≅ (Z + N) mH ≡ AmH, where we have neglected the mass difference between the proton and the neutron, the binding energy of the nucleons (responsible for the cohesion of the nucleus), and the binding energy of electrons to the nucleus (responsible for the cohesion of the atom). The chemical properties of elements are closely related to the atomic number Z, while the physical properties, in which mass plays an important part, are related to the mass number A.

c) Elementary particles and quarks

Particles are usually considered as elementary if they are the smallest part of matter that may be isolated. Besides electrons, protons and neutrons, which are the building blocks of ordinary matter, there are many additional particles, which are observed in cosmic rays or produced in collisions carried out in laboratories using accelerators. Particles are characterized by their mass, charge, spin (intrinsic angular momentum), magnetic moment, etc. A particle may be stable (the electron, the proton, the photon, and the neutrino) or unstable (the free neutron, for instance); in the latter case, they are characterized by their average lifetime ranging from 10−20 s to 898 s for the neutron. Some characteristics of stable particles are listed in Table 1.1.

It is well established that each particle has a corresponding antiparticle of the same mass but opposite charge and some other characteristics. For instance, the positron (e+) is the antiparticle of the electron (e−), the antiproton ( p ) is the antiparticle of the proton (p), etc. The antiparticle and the particle may be identical as in the case of the photon; they are then necessarily neutral. However, the antiparticle of a neutral particle may be different from the particle. This is the case for the antineutron n whose gyromagnetic ratio is opposite to that of the neutron. A particle and its antiparticle may be produced simultaneously in a reaction. For instance, a photon (γ) of sufficiently high energy may be transformed into a pair ch1_eqa17_01 if it collides with a nucleus or another particle N according to the reaction ch1_eqa17_02 Conversely, if a positron encounters an electron, they may annihilate into two photons at least, according to the reaction ch1_eqa17_03 These two reactions are examples of the conservation of the electric charge. No nuclear reaction or particle reaction that violates this law of conservation has ever been observed until now. Thus, it is considered to be a fundamental law of nature. In a process, it is possible to have a transfer of charge, a creation or annihilation of particles of opposite charges, but the total charge of any electrically isolated system is conserved. The total charge of the Universe (which is an isolated system because there is nothing else) is a constant (and probably zero).

Table 1.1. Characteristic quantities of some particles. ch1_eqa17_04 is Bohr's magneton and ch1_eqa17_05 A.m² is the nuclear magneton. ch1_eqa17_06 J.s is Planck's reduced constant. me is the electron mass and mp is the proton mass. The particles behave as small magnets with a magnetic moment parallel to their spin. There are several species of neutrinos

Elementary particles are extremely small and the concept of size is ambiguous at this scale. The electron has an extremely small radius to be measured with present techniques; thus, for all purposes, it is considered to be a point particle. Protons and neutrons have radii of the order of 10−15 m. However, although neutrons are neutral, they have a magnetic moment. Protons are strictly stable, while neutrons may be stable inside the nucleus but, if free, a neutron decay into a proton, an electron, and an antineutrino (beta decay) with a mean lifetime of 898 s. A particle is considered as stable if its mean lifetime is long enough to be observed in a bubble chamber, for instance.

Besides the photon, fundamental particles can be classified into leptons and hadrons. The leptons (including the electron, the muon, the neutrinos, etc.) have electromagnetic and weak interactions but no strong interactions. They are actually considered as strictly elementary. Hadrons (counting about 300 types of particles including nucleons) have all types of interactions. They have a complex structure, so they are not considered elementary, but are comprised of more fundamental entities of charge ±e/3 or ±2e/3, called quarks. However, until now, quarks have never been observed as a separate entity. Thus, the isolated charges are always integer multiples of the elementary charge e. Nucleons are formed by three quarks, but other hadrons are formed by two quarks.

1.9. Interactions in nature

Actually, we know four types of interactions. They can be distinguished by their strength and their range, i.e. the distance over which the forces are significant. They are also characterized by selection rules that we will not consider here.

a) Gravitational interactions

The interaction of two point masses m and M may be expressed by the law of universal attraction F = −GMmr/r³, where G = 6.67 × 10−11 m³/kg.s². The corresponding interaction potential energy UG= GMm/r decreases slowly with the distance, like 1/r. We say that this is a long-range force. As all bodies that have mass have gravitational interaction, this is the dominant force on the cosmic scale. It is responsible for the cohesion of celestial bodies, the binding of satellites to planets, of planets to stars, the stars to galaxies and the galaxies within the Universe.

b) Electromagnetic interactions

These interactions include the Coulomb force between electric charges FE = Koq1q2/r² (where Ko ≈ 9 × 10⁹ N.m²/C²) and the magnetic forces between charges in motion, magnetic matter, and electric currents. These interactions are much more intense than gravitational forces. In the hydrogen atom, for instance, the electrons and the proton are separated by an average distance r = 0.53 × 10−10 m. Their electric attraction is FE = −Koe²/r² = −8.2 × 10−8 N, while their gravitational attraction is only FG = −GmPme/r² = −3.6 × 10−47 N, thus 10³⁹ times weaker. However, the electric forces are rarely perceived on the macroscopic scale, as macroscopic bodies are usually neutral. The Coulomb interaction potential energy is UE = Koq1q2/r, and decreases with distance like 1/r; thus, electromagnetic interactions are long-range forces. The binding energy of particles by electromagnetic forces is of the order of the electron-volt (1 eV = 1.602189 × 10−19 J) and particles that decay by electromagnetic interactions have a mean lifetime of the order of 10−18 to 10−20 s.

c) Strong interactions

These interactions are responsible for the binding of nucleons within nuclei and the binding of quarks within hadrons. They are about 10³ times more intense than electromagnetic forces. Their typical binding energy in nuclei is of the order of 8 MeV per nucleon. The particles, which decay by strong interactions (called resonances), have a mean lifetime of the order of 10−22 to 10−23 s. The strong interactions cannot be formulated as a classical law of force. However, we know that they have a very short range (of the order of the size of the nucleus, i.e. ≈10−¹⁵ m). For this reason, they play no part in atomic and molecular physics (where particles are separated by distances of the order of 10−10 m) in macroscopic physics and in chemistry.

d) Weak interactions

These interactions are responsible for beta decay of the neutron and atomic nuclei and the decay of most of the elementary particles. They are about 10¹² times weaker than electromagnetic interactions, but much more intense than gravitational forces. They have an extremely short range. The particles, which decay by weak interactions, have a mean lifetime of the order of 10−8 to 10−10 s and sometimes much longer if the decay energy is small (for instance, the neutron has a mean lifetime of 898 s).

1.10. Problems

Scalar and vectors

P1.1 Designating the derivatives by primed quantities, show that

P1.2 a) Consider the rotation through an angle φ about Oz. Express the new basis e′α in terms of the basis eβ. Write the transformation equations for the components of a vector field A. Write this transformation in the matrix form A'= RA. What are the transposed matrix ch1_eqa19_02.gif and the inverse matrix ch1_eqa19_03.gif Verify that R is orthogonal. b) Suppose that a magnetic field is given by ch1_eqa19_04.gif Write its expression in the new frame. Considering this rotation, can the expression ch1_eqa19_05.gif be a vector field?

P1.3 To handle complicated vector analysis, it is practical to introduce Levi-Civitta symbols of permutations ch1_eqa19_06.gif . Any permutation (α, β, γ) of (1, 2, 3) may be obtained by successive exchange of indices. We define ch1_eqa19_06.gif as equal to ±1 depending on whether the number of exchanges is even or odd and ch1_eqa19_06.gif = 0 if two indices are the same. Thus ch1_eqa19_06.gif is odd in the exchange of any two indices ( ch1_eqa19_06.gif = ch1_eqa19_07.gif a) Verify that these symbols obey the relation ch1_eqa19_08.gif = ch1_eqa19_09.gif and that the Kronecker symbols obey the contraction relations ch1_eqa19_10.gif . Deduce that the symbols ch1_eqa19_06.gif verify the contraction relations ch1_eqa19_11.gif . The second relation expresses simply that the number of different permutations of (1, 2, 3) is 3! = 6. b) Verify that the determinant of a matrix Mαβ may be written as det(Mαβ) = ch1_eqa19_12.gif and, more generally, ch1_eqa19_13.gif . c) Verify the following relations of the cross product and the triple scalar product

d) Use these Levi-Civitta symbols to calculate the more complicated products:

Integrals involving vectors

P1.4 Calculate the flux of the vector field E = f(r) er through the sphere of center O and radius R. Calculate the divergence of Eand its integral over the enclosed volume and verify Gauss-Ostrogradsky’ theorem.

Gradient and curl, conservative field and scalar potential

P1.5 Verify that the differential operator ∇ is a vector operator. Deduce that the gradient ∇f is a vector, the divergence ∇.A is a scalar and the curl ∇× A is a vector.