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Part I

From classical to modern physics

IOP Publishing

Modern Physics

A critical approach

Canio Noce

Chapter 1

The basic concepts of classical physics as a useful path towards modern physics

Delia Guerra, Ileana Rabuffo and Alfonso Romano

Classical physics includes all the theories formulated after the seventeenth and before the twentieth century, characterized by the implementation of the scientific method proposed by Galileo Galilei [1]. From that moment, nature was investigated by scientists under a completely new perspective. The key feature of classical physics is undoubtedly the determinism which characterizes all its fields, from celestial mechanics to electrodynamics. According to Pierre Simon Laplace, ‘an intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the Universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world … nothing would be uncertain, the future as well as the past would be present to its eyes’ [2].

The birth of quantum mechanics and the theory of relativity in the twentieth century undermined that vision of the Universe and created the crisis of classical physics, highlighting the need to adopt a totally different point of view (see chapters 3, 7 and 8). However, deep knowledge of the concepts of classical physics cannot be avoided if one is to understand modern physics. In fact, a physicist’s mindset is shaped at the beginning when it approaches classic physics. In this chapter, our intent is to present some of the constitutive topics of classical physics, reworked in an educational manner; we propose alternative approaches to the canonical methods, in order to highlight their relevance to students and to prepare them for the study of modern physics. Moreover, the choice of subjects here has been made by considering each of them as preparatory and useful for the comprehension of the subsequent subjects.

In section 1.1 we introduce the three principles of dynamics with the purpose of underlining their conceptual value and their revolutionary importance. Section 1.2 deals with the concepts of work and energy, putting in evidence the strong correlation between these two quantities; in addition, profound knowledge of the classical concept of energy allows its quantization in quantum mechanics to be understood even more. This motivation, namely the ‘quantum jump’ which characterizes certain physical quantities, is also the reason why section 1.3 is devoted to the introduction of the momentum of a vector. In particular, angular momentum is generally a difficult concept for students; therefore we propose a pedagogical path in order to make it easier to understand. Section 1.4 is dedicated to a fundamental argument, which is hardly ever taken into account in basic physics courses, despite its unquestionable relevance: the link between symmetries and conservation laws, as investigated within the celebrated Noether theorem. Finally, sections 1.5 and 1.6 are devoted to the concept of the wave and to Maxwell’s equations, respectively, showing how it is possible to teach students that an undulatory behaviour for coupled electric and magnetic fields can, under suitable conditions, be predicted. We propose an alternative way to present this topic by introducing the concepts of divergence and curl of a vectorial field in a simplified but correct perspective.

It should be noted that in this context we mostly deal with mechanics rather than thermodynamics or electrodynamics. Actually, we believe that stressing these subjects is significant, since by learning to handle classical mechanics it is possible to acquire an universal method which turns out to be useful in every field of physics (and not only in physics) [3].

1.1 The Newton principles of dynamics

1.1.1 The principle of relativity and the first principle

According to Aristotle, the natural state of bodies is rest; thus, any body in movement tends to get slower and slower until it stops, unless it is pushed to continue its motion. Aristotle reached this conclusion, which is evidently wrong, based on his experiments. On the other hand, Galileo interpreted the same experiments in a completely different way: if the object tends to slow down then there is a force opposed to its motion, called the force of friction. He recognized (for the first time) the rank of ‘force’ for the phenomenon of friction. The difference between the two approaches is undoubtedly the scientific method used by Galileo; through it, the Aristotelian false belief about motion was corrected and the results can be summarized in the following statements:

If two laboratories move with uniform linear motion, no experiment will give different results in either of them.

A reference system moving with uniform linear motion is defined as inertial.

In an inertial reference system, a physical object preserves its rest or its uniform linear motion if it is not forced to change this state by the application of an external force.

The first point is the well-known principle of relativity. Point 3 is the principle of inertia, formulated by referring to the definition of an inertial reference system given in point 2. It should be noted that ‘force’ is intended here as a ‘real force’, i.e. an interaction due to something with physical relevance. A fictitious force, in contrast, is the product of the mass of the object through a non-inertial contribution to the acceleration that may emerge in the reference system in which it is located.

The first principle has revolutionary value: it states that the state of rest and the state of uniform linear motion are indistinguishable. This bold assertion is not simple to accept, since it is contrary to common intuition. The first principle also has a fundamental conceptual value, since it allows a special class of reference systems to be defined, that is, the inertial systems.

However, it also has a logical irregularity: according to its formulation, in order to know if one reference system is inertial we have to assure that no forces act on an object in it. Then, we have to verify the state of motion of the object: if it is at rest or moves with a constant speed, the reference system is inertial, otherwise, if it moves with a finite acceleration the reference system is not inertial. However, this demands certainty that no real force is acting on it, but this is formally impossible, since we do not have a quantitative definition of what we mean when we say that a force is zero. We can use the first principle ‘back to front’, namely as a principle allowing us to deduce whether a net force acts on an object. This, however, requires knowing that one is in an inertial system, so that one definitely ends up with a circular definition. In order to overcome this difficulty, it is assumed that an inertial reference system by definition exists, having its origin in the Sun and its axes oriented toward the fixed stars. As a consequence, all reference systems which are at rest or which move with constant speed with respect to this inertial system are inertial as well. In this way, once an inertial reference system is selected, one can immediately establish if some net forces act on a given object, by checking if it is at rest or in a state of uniform linear motion. It is clear that, under these conditions, the first principle can be used as a criterion to verify the presence or the absence of forces.

1.1.2 The second principle

The second principle deals with the concept of force. It was formulated by Isaac Newton who had in mind real forces only. There are several misconceptions about forces, which can be pointed out by asking students the following questions:

Do we need contact to exert a force?

Does the application of a force move an object?

If an object moves under the action of a force, does it always move in the same direction of the applied force?

Everyday experience leads to incorrect answers to these questions, in particular to the second one: students are tempted to answer ‘yes’ since they confuse the concept of force with the concept of energy: work always implies motion, whereas the same cannot be said for a force.

The first misconception can be immediately removed if we refer to well-known examples, such as the gravitational force or the electrostatic interaction, which demonstrate that contact is not necessary to exert a force on an object.

The answer to the second question can be suggested by pushing a desk and observing that it does not move if the applied force is not sufficient to overcome the maximum static friction force. Thus, it is evident to students that the application of a force does not always generate motion.

The third point is probably the most subtle, as one can understand on the basis of the following example: there is an attractive force between the Earth and the Moon and it is directed along the line which links their centres, however, the Moon does not move toward the centre of the Earth. Why does this happen? The answer is that the application of a force makes a body acquire an acceleration which is always parallel to the force, but this does not imply that its displacement, i.e. its velocity, is parallel to the force too.

The innovative significance of the second principle is that it establishes the real relation between the cause of the motion (a force F⃗ doing finite work) and its effect (the acceleration a⃗). Since it predicts what happens when causes are known, it introduces the profound concept of determinism, which is important also from a philosophical point of view: specifying position and speed at a given time, the equation F⃗=ma⃗ gives the possibility of describing the motion of an object completely. This point is crucial and characterizes classical mechanics, differently to what happens in quantum mechanics. The second principle has also an important conceptual significance, since it allows the mass of an object to be defined, beyond Newton’s idea that it simply denotes the quantity of matter. The second principle refines this interpretation and defines the mass as the ratio between the modulus of the net force applied to an object and the modulus of the acceleration that it acquires. This ratio identifies the so-called inertial mass of an object, which is different to the gravitational mass, i.e. the property which we measure with a weight scale. Nevertheless, it is proved rigorously that for the same object the two values are the same.

In the following, we report three more widespread student beliefs that turn out to be false.

Students often believe that gravity does not act in a vacuum; therefore, for example, in their opinion astronauts float in a spaceship because they are in a vacuum. This is of course not true: this actually happens because the spaceship is subject to gravity in exactly the same way as the astronauts are, in the sense that they are both in free fall. In order to clarify this concept, let us consider a lift and a student in it. For an inertial observer, we have that the projection of F⃗=ma⃗ along the vertical direction leads to

mg−N−T=ma,(1.1)

where N is the normal force, T is the tension of the cable, a is the acceleration of the lift and g=GMTr2 is the acceleration of gravity (G is the gravitational constant, MT is the mass of Earth and r is the distance of the student from the centre of the Earth). Then, from equation (1.1) it follows that:

N=mg−ma−T=m(g−a)−T.

If the cable of the lift breaks (T = 0), then the lift is in free fall; this means that a = g and, as a consequence, N = 0, i.e. the previous contact between the student’s feet and the lift floor ceases to exist (see figure 1.1). The student, then, begins to float, since he is falling and the lift is falling in exactly the same way. This is what happens to the astronauts in the spaceship, namely the astronauts and the spaceship are together in free fall around the Earth.

In order to point out a common mistake, sometimes reported in considerations of the physics of sport, one can consider the case of a ball hit by a racket; in the study of the motion of the ball, the force that the racket exerts on it must not be included in the equation F⃗=ma⃗, because this force only affects the initial conditions of the motion, determining the value of the initial velocity ₀. More precisely, we deal in this case with an impulsive force acting during a very short time interval Δt, which causes a variation of the quantity of motion in one dimension, given by

Δp=mv0−0=FΔt.

Consider a pendulum and the projection of F⃗=ma⃗ in the direction of the cable; in some cases students are tempted to write

T−mgcosθ=0

since there is no motion along the cable. Obviously, this is wrong since the normal component of the acceleration is clearly non-vanishing.

[picture]

Figure 1.1. (a) A student in a lift going down and (b) the same system in free fall when the cable is broken.

1.1.2.1 Most common errors

In this subsection, we discuss some common mistakes made by students in the application of the second principle of dynamics. This aspect should be taken into consideration: it is important to show students what should be done and what should not. In our opinion, students should be encouraged in particular when they get something wrong, since errors can be useful in order to obtain complete comprehension. In the left panel of figure 1.2, we show a mistake in writing the second law of dynamics, while the panels in the centre and on the right relate to two widespread errors with regard to the formal definition of the force of friction and the acceleration of an object on an inclined plane, respectively. Another misleading belief of students is that the so-called centrifugal force is an active force pushing an object out of its trajectory. This is obviously false, since it is a fictitious force, i.e. not due to a real interaction, that appears to act on an object when viewed in a rotating frame of reference. Some confusion may also arise with the concept of centripetal force, which students in some cases consider as a specific applied force. In contrast, the real physical quantity to refer to is the centripetal acceleration: when it is non-vanishing, then the product of mass for acceleration is the force, which sometimes is a tension, sometimes a reaction constraint, and so on.

[picture]

Figure 1.2. Serious mistakes in writing the second law of dynamics (left), the force of friction (centre) and the force of gravity on a inclined plane (right).

Generally speaking, the most frequent errors are related to the use of vectors, which apparently often causes difficulties for students. In figures 1.3 and 1.4, we report other mistakes, in this case concerning the scalar and the vectorial products and the definition of pressure, respectively. In particular, the image on the right in figure 1.4 shows a double mistake: the first is, of course, that pressure is not a vector; the second one deals with the fact that, in vector algebra, the inverse of a vector is not defined (note that this mistake is also present in the image on the left of figure 1.2).

[picture]

Figure 1.3. Two mistakes for the scalar and the vectorial product.

[picture]

Figure 1.4. Incorrect relations defining pressure.

1.1.3 The third principle

An important aspect of the third principle is that opposite forces between two particles act along the same straight line. Without this clarification, in several cases students could be misled. For example, by considering an isolated system with two particles, the conservation of the linear momentum gives:

where F⃗1 is the force that m₂ exerts on m₁ and F⃗2 is the force that m₁ exerts on m₂. One could thus be tempted to conclude that the third principle is a consequence of the second one, but this is not true since equation (1.2) does not imply that F⃗1 and F⃗2 act along the same line. A possible suggestion to give to students is that every time we individuate a force acting on an object, there is always its partner, as required by the third principle; however, it is important to remember that the correct partner is not included in the equation of motion of the object we are interested in. A check on this point could be to ask a student: ‘Consider an object lying on a table. Is the normal force exerted by the table on the object the partner of the weight on the object?’. Frequently the student answers ‘yes’, but this is the wrong answer, because the partner is the force exerted by the object on the Earth.

1.2 Work and energy

1.2.1 The concept of work

The concept of work is a central one, since it is strictly related to the concepts of kinetic and potential energy. This is of course of special relevance, since every form of energy (thermal, electric or nuclear) is always attributable to the forms of kinetic and potential energies of the elementary constituents. The importance of the concept of work is thus undoubted, thus it should be introduced in basic mechanics courses in a careful and appropriate way. The fundamental definition of work should be referred to the simplest situation, i.e. constant force and linear displacement; under these conditions, it is defined as the product between the force and the displacement which the force generates in the direction of the force itself. This means that all the forces giving rise to displacements orthogonal to their own direction do not produce work. This is what happens, for instance, in the case of a force acting on a particle which moves in a uniform circular motion. In contrast, if the force and the displacement are in the same direction, the work is maximum: this is what happens, for example, when a force pushes an object on a constraint in the direction parallel to the constraint. An elegant way to express this concept is the use of the scalar product:

L=F⃗·Δs⃗=FΔscosθ.(1.3)

Here θ is the angle formed by the direction of the force F⃗ and the direction of the displacement Δs⃗. This definition, which, as already stated, refers to the special case of constant force and linear displacement, can be immediately used to give the definition of work in the general case of variable forces and curvilinear trajectories. It is sufficient to divide a given trajectory C into a series of elements small enough that they can be considered linear and that the force along each of them can be assumed to be approximately constant. The general procedure is illustrated in figure 1.5, where the red vectors represent generic displacements Δs⃗i (i=1,2,3,…) along the elements into which the trajectory has been divided. Under the conditions specified above, the definition (1.3) can be applied to each linear element, so that one can introduce the sum

F⃗1·Δs⃗1+F⃗2·Δs⃗2+⋯+F⃗i·Δs⃗i+⋯=∑iF⃗i·Δs⃗i.

Work is then defined as the limit of this quantity when the number of elements into which the trajectory has been divided becomes arbitrarily large or, equivalently, the length of each them becomes arbitrarily small:

L=limΔs⃗i→0∑iF⃗i·Δs⃗i=∫CF⃗·ds⃗.(1.4)

The integral in equation (1.4) refers to a calculation along a line; indeed, the subscript C denotes that the result of the integral is intimately connected to the path along which the calculation is performed. This is a point to stress with students, since the symbol ∫ is also used to indicate ordinary integrals, defined, according to figure 1.6, as

∫abFxdx=limΔxi→0∑iFξiΔxi.(1.5)

It is evident that whereas the above quantity is only related to the dependence of F on x, in equation (1.4) the result is also dependent on the path C. Thus, equations (1.4) and (1.5) are profoundly different in their significance. Integrals such as the one in equation (1.4) are called line integrals to point out the importance of the path for their evaluation, while integrals such as that in equation (1.5) are the ordinary Riemann integrals. We also point out that when the line integral of a given vector is calculated around a loop, the integral is often called circulation of the vector.

[picture]

Figure 1.5. Illustration of the calculation of a line integral.

[picture]

Figure 1.6. Illustration of the calculation of a definite integral.

In this context, it is worth mentioning a special case: the work done by a fluid as a consequence of variations of its volume, as usually considered in thermodynamics. The line integral, in this case, becomes an ordinary integral and the work reads

L=∫VAVBPdV,

where VA and VB are the values of V in the initial and final equilibrium states A and B, respectively. In the above expression, P=F⊥/A is the pressure of the fluid, with F⊥ being the force orthogonal to its surface and A is the section of the recipient which contains the fluid, taken parallel to its surface. This integral is graphically represented in figure 1.7.

[picture]

Figure 1.7. Representation of the thermodynamic work in the P–V plane.

Line integrals are a widely used mathematical tool. The definitions of the electric potential difference and the Ampère circuital law, for example, are expressed in terms of line integrals:

VB−VA=−∫ABE⃗·ds⃗∮B⃗·ds⃗=μ0i.

Going back to equation (1.4), it is evident that the value of a line integral is a function of the chosen path. However, there exists a special category of forces, called conservative forces, for which the line integral is independent of the path and only depends on its initial and final points. In these special cases, one introduces a function V which allows the result of the line integral to be written in the form

∫CF⃗·ds⃗=V(f)−V(i),(1.6)

whatever the line C connecting the initial and the final points i and f, such that the integral only depends on i and f. In addition, if the initial point i is kept fixed, equation (1.6) allows each point of the space to associate to a number, that is, the work done by the conservative force when a particle moves from i to that point (we deal with this argument in more detail in the next section). This also implies that if we choose a closed line, the value of the integral is always zero.

1.2.2 The concept of kinetic energy

Kinetic energy is defined as the energy of an object of mass m in movement with speed equal to and it is expressed as . It can be useful to motivate such a definition by showing students its origin and the important link with the concept of work illustrated in the previous section.

Let us consider a particle of mass m and all the forces acting on it. If we define F⃗ as the sum of all these forces, the second principle of dynamics asserts that F⃗=ma⃗, where a⃗ is the acceleration of the particle. Then, we can calculate the work L done by F⃗ relative to a displacement from point A to point B:

Here, starting from , we have used , so that . The calculation in equation (1.7), which expresses the so-called work–energy theorem, shows that the work done by all forces acting on a particle equals the variation of a quantity, , depending on the modulus of the velocity, defined as the kinetic energy of the particle. The above result is also valid in special relativity [4]. Writing the force as

and calculating the derivative explicitly, we obtain

Given this expression of the relativistic force F⃗, we can evaluate the elementary work dL. After some simple algebra, we obtain

This expression can also be written in the form

We can now evaluate the finite work L done by F⃗ when a particle moves from A to B by integrating equation (1.8):

This result allows the relativistic kinetic energy to be defined as

where K₀ is a constant. By imposing the condition one immediately finds K0=−mc2, so that the final expression of the relativistic kinetic energy is

It is worth noting that, expanding this expression in Taylor series for and retaining terms up to second order, one recovers the classical expression . The above arguments thus prove that the concept of kinetic energy is intimately related to the concept of work also in a relativistic frame.

We now show that the arguments treated above can also be presented without the use of the line integral. Of course, in what follows there are simplifying hypotheses about the trajectories and the way in which an object travels along them; nonetheless this approach is worth discussing, since the line integral is a mathematical tool that students generally are not familiar with.

Suppose that a particle of mass m moves from A to B along a curved trajectory (see figure 1.8) under the action of a resulting force F⃗; for simplicity, we will consider a linear displacement Δs⃗ from A to B and we will suppose that F⃗ remains constant along this linear path. Then, the work done by F⃗ is given by

Since we are considering a linear displacement, vectors and are parallel; then, and we write

In this equation, the speed is intended as the average speed , then

Equation (1.9) is exactly the same theorem of kinetic energy obtained in the previous section (see equation (1.7)). The difference here is that we have obtained the same result with a formally easier mathematical approach.

[picture]

Figure 1.8. A curved trajectory from point A to point B (blue line) and the corresponding linear displacement (red line).

1.2.3 The concept of potential energy and the principle of conservation of mechanical energy

We observe that in the most general case the resultant force includes two different categories of forces, conservative and non-conservative forces. As already mentioned in the previous section, the work done by a conservative force does not depend on the trajectory along which a particle moves. Looking at figure 1.8, this means that for the determination of its value, it is not important if the displacement of the particle from A to B is along a straight line or along a curve—conservative forces do the same work in the two cases. This implies, as already mentioned in the previous section, that if we keep point A fixed, we can associate a number with the point B, V(B), equal to the work done by the conservative force when the particle moves from A to B. Generalizing this procedure, we can associate with each point of the space (C, D, E …) a number (V(C), V(D), V(E) …) representing the work done by the conservative force when the particle moves from A to that point. In this way, we have obtained a mapping of the space that associates with each point a number having the dimensions of an energy (as such measured in joules). Reversing the sign of this quantity, we obtain the potential energy U that a particle has when it is at that point. It is called ‘potential’ since it refers to an energy intended ‘to be spent’. In this way conservative forces give to the points of the space a kind of energetic classification. Therefore, the work done by a conservative force when a particle moves from point A to point B can be written as

L=−[U(B)−U(A)].(1.10)

Positive work done by the force thus implies a displacement accompanied by a decrease of the potential energy.

We have not justified so far the use of the term ‘conservative’. What is conserved? We can rewrite the work–energy theorem separating in L the work Lcons done by conservative forces from the work Lnon‐cons done by non-conservative forces:

Then, using equation (1.10) one has

Rewriting this equation in the form

one obtains an expression for the work done by non-conservative forces. Note that in the absence of non-conservative forces, the quantity

defined as the mechanical energy of the particle at a given point P, remains unchanged when the particle moves from A to B. In this case, the kinetic energy and potential energy both vary in time along the trajectory, but always in such a way that their sum stays constant. This is the reason why forces doing work not dependent on the trajectory are called conservative—when they are the only kind of forces acting on a particle, the mechanical energy defined in equation (1.12) is a conserved quantity. This conservation law is of great importance in mechanics since it entails a constraint on the behaviour of the particle. During its movement it can change speed (i.e. the kinetic energy) and position (i.e. the potential energy), but cannot change their sum. If conservative and non-conservative forces are simultaneously present, then equation (1.11) can be seen as an energy balance equation, since it states that the energy loss, which is the work done by non-conservative forces, is the difference between the final and the initial values of the mechanical energy. Obviously, this difference is always negative.

1.3 Angular momentum

It is well known that, in order to represent a vectorial quantity, we use arrows. In many cases an arrow can be moved in space without changing its magnitude and direction since this operation of ‘parallel transport’ does not change its meaning. Vectors of this type are, for example, force, velocity, acceleration and so on. For these quantities, a single arrow represents a class of equivalence. However, other physical quantities are associated with vectors which cannot be translated in space without consequences, since their application at a point has a different effect with respect to their application at another point. For this reason, the definition of these vectors is always given in combination with a position vector r⃗ specifying their point of application in space. It follows that in order to define the action of an applied vector A⃗, we have to assign a second vector r⃗, specifying the point of application P with respect to an origin O (figure 1.9). In particular, one can combine these two vectors by means of the operation of the vectorial product, in this way defining a fundamental quantity in physics, which is the momentum of A⃗ with respect to the point O:

M⃗=r⃗×A⃗.(1.13)

It is orthogonal to the plane which contains r⃗ and A⃗ and retains memory of the relative position and orientation of the two vectors. The momentum has two relevant properties:

It depends on the position from which we see it—by changing O the vectorial product equation (1.13) changes correspondingly.

It does not change when A⃗ is moved rigidly along the straight line corresponding to its direction.

There are two fundamental quantities which are represented by vectors of this kind: the momentum of a force, or torque, and the momentum of the quantity of motion, or angular momentum.

[picture]

Figure 1.9. Schematic illustration of the applied vector A⃗ and the position vector r⃗ specifying the point of application P of A⃗ with respect to point O.

The momentum of a force, called torque, is defined as M⃗=r⃗×F⃗. In order to understand its role in mechanics, we can consider the case in which F⃗ changes its application point in space and r⃗, which follows it, does not change in magnitude. As is evident from figure 1.10, in this case the point of application of F⃗ moves on a circumference. Then, if F⃗ is applied to a mass m, regardless of whether we are considering a particle or a rigid body, the effect produced by F⃗ is a rotation.

[picture]

Figure 1.10. Schematic illustration of a vector F⃗ (red arrow) whose application point changes. The vector r⃗ (blue arrow) giving the position with respect to O of the application point is assumed to have a constant modulus.

The same arguments can be used for the definition of angular momentum, given by

Here we face the conceptual difficulty that L⃗ is a physical quantity not related to the causes of motion. Rather, it is intrinsic to the behaviour of the particle and thus its significance is not immediately evident. Assuming, for simplicity, that r⃗ does not change in magnitude, the particle describes arcs of circumference, so that the magnitude of its quantity of motion is equal to , where ω is its angular velocity. In addition, the vector has to remain orthogonal to r⃗ and then the magnitude of the angular momentum is

The direction of L⃗ is orthogonal to the plane which contains r⃗ and , with its orientation directly related, via the right-hand rule, to the direction of rotation. In this way, the angular momentum keeps track of everything dealing with rotation.

More interesting is the extension of the above considerations to the case of a system of particles and, in particular, to a rigid body. Suppose we have a symmetrical rigid body, treated for simplicity as a discrete set of particles, and we put it in rotation around one axis of symmetry. In this case, schematically shown in figure 1.11, a point of the rigid body with mass mi and located at a distance ri from the axis rotates on a circumference of radius ri with an angular velocity ω which is the same for all particles in the body. Then, for every selected point in the body there is another point located symmetrically with respect to the rotation axis, so that the related contributions to the total angular moment L⃗ have components that cancel out in the direction perpendicular to the axis and sum up along the axis itself. Then, the magnitude of the angular momentum is given by the sum of many contributions, each having the form of equation (1.14). Defining the angular velocity as a vector ω⃗ of magnitude ω and the direction of the rotation axis, oriented in such a way as to see the rotation taking place counter-clockwise, we can say that the direction of L⃗ is in this case the same as that of ω⃗. Thus we can write

L⃗=∑imiri2ω⃗=Iω⃗.(1.15)

The scalar quantity I=∑imiri2, defined as ‘the moment of inertia’, gives information not only on the mass, but also on the way it is distributed around the axis of rotation, which is a relevant point. From equation (1.15) one can easily understand why L⃗ is also called the momentum of the quantity of motion—the analogy between this equation and the definition of the quantity of motion is evident and allows us to guess that the quantity I is a measure of the inertia that a rigid body offers to rotations, exactly in the same way as the mass is a measure of the inertia that a particle offers to translations. However, it should be noted that equation (1.15) is valid only if the rotation takes place around a symmetry axis of the rigid body. When this condition is not realized, L⃗ is not in the direction of ω⃗, but nonetheless for its axial component L∥ one can write a scalar relation reminiscent of equation (1.15):

L∥=Iω.

The above considerations are of great relevance, since they allow students to be shown an interesting analogy between the translational motion, governed by F⃗=ma⃗, and the rotational one, governed by the momentum of the forces through an equation which can be immediately obtained by deriving equation (1.15) with respect to time (again considering rotations around a symmetry axis):

M⃗=Iα⃗.(1.16)

Here M⃗ is the resultant momentum of the external forces with respect to a pole lying on the rotation axis, I is the moment of inertia with respect to the same axis and α⃗ is the angular acceleration taking into account variations of the angular velocity. These quantities can be seen as the rotational analogue of the total force F⃗, the mass m and the linear acceleration a⃗, respectively. In this context, it is important to note that rotational dynamics is affected not only by the value of the mass of the rotating body, but also by its distribution around the rotation axis.

[picture]

Figure 1.11. A symmetric rigid body rotating around its symmetry axis.

When the angular momentum is not parallel to ω⃗, namely it is not along the axis of rotation, but has also a transverse component L⃗⊥ (L⃗⊥=Lxxˆ+Lyyˆ, if the z-axis coincides with the axis of rotation) then an additional term appears in the equation of the rotational motion:

M⃗=Iα⃗+dL⃗⊥dt.

Separating axial and transverse contributions according to

M⃗∥=Iα⃗M⃗⊥=dL⃗⊥dt,

we see that M⃗∥ and M⃗⊥ are responsible for variations of the angular momentum along the axis of rotation and orthogonally to this axis, respectively. It is evident that when M⃗⊥=0 we go back to equation (1.16). In this case a motion with constant angular velocity can take place with no need to apply external momenta. Equivalently, we can say that the most efficient way to put a rigid body in rotation is to make this happen around an axis of symmetry.

The analysis can be made more general, observing that for every rigid body it is always possible to find three orthogonal axes such that in a rotation around one of them, the direction of the angular momentum would coincide with the one individuated by that particular axis. We can thus say that if aˆ, bˆ and cˆ are the unit vectors of these ‘special’ axes, then in analogy with equation (1.16) the total angular momentum is L⃗a=Iaωaˆ if the body rotates around the aˆ-axis, is L⃗b=Ibωbˆ if it rotates around the bˆ-axis and is L⃗c=Icωcˆ if it rotates around the cˆ-axis. The scalar quantities Ia, Ib and Ic give information about the distribution of the mass with respect to the three axes and are called central moments of inertia. Each of them has the form ∑imiri2, where ri are the distances from the specific axis we are considering of the particles making up the rigid body (in a discretised description). When the rotation is around a generic axis, the relation connecting the angular momentum to the angular velocity is more complicated and takes the matrix form

L⃗=Iω⃗,

where I is a 3 × 3 matrix, known as the tensor of inertia. We thus see that different to the case of rotation around a symmetry axis, each component of L⃗ is now expressed as a linear combination of all three components of ω⃗.

1.4 Symmetries and conservation laws

Conservation laws are of great relevance in physics since they capture some general regularities of Nature. Even though their deduction is based on the second and the third principles of dynamics, such that they have the same ‘information content’ as the Newtonian laws, they allow a deeper understanding of the behaviour of a dynamic system, at the same time providing a powerful tool for the solution of specific problems.

The three fundamental conservation laws of mechanics, which concern mechanical energy, quantity of motion and angular momentum, have never been violated by any experiment or theory. They hold in quantum mechanics as well, where their role is even more crucial than in classical mechanics or relativity, given that they often represent a unique instrument allowing us to understand microscopic phenomena. Therefore, conservation laws are universal and control all physical processes directly—if they do not respect these laws, they are forbidden. Conversely, if a predicted event respects conservation laws, this could certainly be verified by means of a suitable experiment. In this sense, we could say that conservation laws delimit what can happen from what cannot.

The reason why conservation laws are inviolable lies in their direct link to symmetry principles, as established by Noether’s theorem. In general, if something is symmetrical it means that it shows an invariance with respect to a change of the point of view, that is, it does not change under certain space transformations. For example, the object in figure 1.12(a) is invariant if we rotate it by 120° with respect to the centre (discrete symmetry), whereas the object of figure 1.12(b) is invariant under rotation by every possible angle (continuous symmetry). For a given symmetry, there is a transformation of coordinates leaving unchanged the equations which describe the dynamics of the system under consideration. The technical definition of symmetry is indeed invariance under certain transformations. Among all the transformations which can be applied to the equations of motion in any context, three of them play a special role:

Translation of the axis of time.

Translation of the origin (implying a change of position).

Rotation of the axes (implying a change orientations).

Symmetry means that these transformations leave the equations of motion unchanged, this being associated, respectively, with:

Homogeneity of time.

Homogeneity of space.

Isotropy of space.

Emmy Noether demonstrated that a conservation law stems from each of these fundamental symmetries, where the concept of symmetry is intended as invariance with respect to a continuous coordinate transformation. The physical quantities conserved correspondingly are:

Mechanical energy.

Quantity of motion.

Angular momentum.

[picture]

Figure 1.12. Schematic representation of objects characterized by certain symmetries.

1.5 A brief description of waves

1.5.1 General remarks

A wave is a disturbance which travels in space and time. More specifically, wave motion is a perturbation on a microscopic scale which moves on a macroscopic scale. Waves carry energy and linear momentum, but not matter. This implies that the material through which they move is stationary. A stone thrown into a lake can intuitively give the idea of the propagation of a wave: a series of concentric rings is generated, and they travel away from the point at which the stone fell into the water. In this case, the physical quantity which is perturbed is the position of the particles of the liquid which oscillate around their equilibrium position. We can consider the wave as the propagation of a state of motion in matter. This is the way, for instance, in which sound or an earthquake propagate (but there is also a kind of wave which does not need matter to propagate, the electromagnetic wave). If the material is isotropic and there are no obstacles, the wave front is spheric and the corresponding wave is a spheric wave. If we observe the wave at a point which is far away from the source that generates it, the portion of the wave front is assimilable to a plane—in this case, we deal with a plane wave. The evolution of a wave depends on the physical properties of the material in which the wave propagates, in particular on the nature of the boosting forces acting on the particles of the perturbed fluid. The simplest case is when the force is an elastic one, then the wave is an elastic wave.

We can classify waves on the basis of the direction of movement of the individual particles of the medium relative to the direction of propagation of the wave itself. According to this, in an elastic medium there are two kinds of waves: longitudinal waves, which are characterized by the fact that the particles move in the same direction as the propagation of the wave, and transverse waves, in which particles move in a direction which is orthogonal to the direction of propagation. For example, sound waves are longitudinal waves, while electromagnetic waves are an example of transverse waves (although, as mentioned above, in this case wave propagation does not involve matter). Figure 1.13 presents a schematic illustration of the propagation of a longitudinal wave in a gas and of a transverse wave on a rope.

[picture]

Figure 1.13. Schematic illustration of how a longitudinal wave in a gas and a transverse wave on a rope can be generated. In the left panel the source moves left and right, while in the right panel the source moves up and down.

1.5.2 Mathematical description

According to general considerations, we can give a mathematical description of a wave starting from the following expression:

Here α is the physical quantity whose perturbation propagates along the x-axis. It is mathematically described by a function depending on space and time through the combination , where is the speed of the wave. This feature is the hallmark of undulatory behaviour. We distinguish between progressive waves, i.e. waves propagating in the positive direction of the x-axis, and waves propagating in the opposite direction. They are described by the functions and , respectively.

Common sense always associates a sinusoidal curve with the concept of a wave, however, this is not necessarily so. Only when the source of the wave is an oscillating disturbance, the correspondent wave is a harmonic one, described by a sinusoidal function:

A classical example of a propagating wave refers to a string with an oscillating extremity, the other end being fixed. The disturbance propagates along the string with transverse oscillation and every point located at x is at time t at a height y given by equation (1.18). A graphic of y as a function of t for fixed x, and as a function of x for fixed t, is presented in figure 1.14. Looking at equation (1.18), we see that the constant A is the maximum value which y can assume and thus is called the amplitude of the wave. The fact that A does not depend on x and t means that there is no dissipation, namely the wave does not decay in its propagation. The quantity k is inserted as a constant making the argument of the sine in equation (1.18) adimensional, but it also has a relevant physical meaning. By writing equation (1.18) in the form

it is easy to see that, for a given value of t, when kx=2πn with n being an integer, the function in equation (1.19) repeats itself in space. This means that all points x=2πkn distant from each other are in ‘spatial phase’, and thus the quantity

λ=2πk,(1.20)

called the wavelength, represents the spatial period of the wave. As a consequence, k=2πλ is the spatial frequency of a wave, i.e. the number of wavelengths per unit distance, for this reason k is called the wave number.

[picture]

Figure 1.14. Evolution in time (left) and space (right) of a wave.

On the other hand, is linked to the temporal periodicity of the wave. For a fixed value of x, one has that when , n being again an integer, the function (1.19) repeats itself in time. Then, waiting for a time equal to an integer multiple of

the wave resumes its value at the chosen point x. Equation (1.21) defines the temporal period T of the wave. One also defines the frequency ν of the wave as

This can be interpreted as the number of waves that pass through a given point per unit time. It is also useful to introduce the angular frequency ω, defined as

ω=2πν.

Then, taking into account equations (1.22) and (1.20), we have

From the previous equations, we also obtain

This relation is important since it connects the spatial and the temporal periodicity of the wave via its speed, this being at the origin of the undulatory behaviour. The characteristic parameters defined above are represented in figure 1.15, where a wave is plotted as a function of the space variable x at a fixed time (top panel), and as a function of time for a given value x (bottom panel). We also remark that when two waves move simultaneously in a medium, they give rise to a new wave, according to the so-called superposition principle. On the basis of the above considerations, the equation that the function (1.17) must obey in order to correctly describe a wave has to take into account the following points:

A wave is always described in terms of a function of space and time variables appearing in the special combination

When two waves move simultaneously in a medium, they form a new wave, according to the so-called superposition principle.

This implies that the equation we are looking for has to treat the coordinate x and the product (point 1) on an equal footing and it has to be linear (point 2).

[picture]

Figure 1.15. Spatial (top) and time (bottom) evolution of a wave with its characteristic parameters.

We propose a heuristic approach to derive this equation, limiting ourselves for simplicity to waves propagating in one dimension with speed equal to . First of all, the variation of the function f with respect to x is expected to appear in the equation in the same form as the variation of f with respect to does. A starting point may be

Equation (1.23) is relative to finite variations. In order to extend equation (1.23) to a continuum, it has to be written using derivatives. In particular, since f is a function of more variables, we have to introduce partial derivatives and thus we could tentatively write

Here the symbol ∂ denotes a partial derivative, performed with respect to a given variable and considering the others as if they were constant quantities. Equation (1.24) is, however, not correct, since it does not treat the dependence on and on the same footing as it should. A possible way to overcome this difficulty could be to square equation (1.24), but in this way we would lose the linearity of the equation. The problem is settled if we consider an equation of the form (1.23), but involving second derivatives rather than first derivatives:

This is exactly the equation of waves known as the d’Alembert equation, which is a second-order linear partial differential equation. It is easy to verify that a solution of equation (1.25) is given by the sinusoidal function defined in equation (1.18). Solutions of equation (1.25) satisfy the superposition principle, according to which a linear combination of solutions of the equation is again a solution. Even though the derivation presented above strictly refers to waves which propagate in one dimension, nevertheless a similar description holds when waves in more dimensions are considered.

1.5.3 Interference and diffraction

The superposition principle has fundamental implications as far as wave propagation is concerned. Suppose we have two waves with the same amplitude A, the same wavelength λ and the same frequency ν, described by the functions

y1(x,t)=Asin(kx−ωt+φ)(1.26)

y2(x,t)=Asin(kx−ωt),(1.27)

with φ being the so-called phase constant. A finite value of φ gives rise to the behaviour represented in figure 1.16, where one of the two waves is shifted with respect to the other. By applying the superposition principle, the wave emerging from the overlap of equations (1.26) and (1.27) is

y(x,t)=y1(x,t)+y2(x,t)=2Acosφ2sinkx−ωt+φ2,(1.28)

where we have used the prostapheresis formula. Equation (1.28) describes a wave which has the same wavelength and frequency as the original waves and has an amplitude which is not the sum of the amplitudes of the wave equations (1.26) and (1.27). A common misconception! This is just the peculiar phenomenon called interference. In particular, we observe that the amplitude of the wave described by equation (1.28),

A0=2Acosφ2,

is equal to the sum of the amplitudes of equations (1.26) and (1.27) only in the particular case in which φ=0 or, in general, when φ=2nπ, n being an integer. This is the case of constructive interference. In contrast, if φ=π/2 or, more generally, φ=2π(n+12), the total amplitude is zero and we have destructive interference. This phenomenon is of great relevance and characterizes the wave behaviour in a unique way—we generally refer to it to recognize undulatory propagation in nature.

[picture]

Figure 1.16. Waves out of phase.

In addition, interference gives rise to another very relevant phenomenon, typical of wave propagation—diffraction. It occurs when a wave encounters an obstacle or a slit, beyond which secondary waves are generated in different directions with respect to the original incidence wave. Since these waves are characterized by path differences, we have interference between them that at some points is constructive and at other points destructive. This behaviour generates the characteristic patterns on a screen, called diffraction patterns, such as those shown in figure 1.17. Their distinctive feature is the alternation of bright and dark zones corresponding to constructive and destructive interference, respectively. Note that diffraction is an interference of a wave with itself which develops only if λ∼d, where λ is the wavelength of the wave and d represents the dimension of the slit. Conversely, when λ≫d, the wave moves undisturbed, as if the obstacle was not there, whereas if λ≪d, undulatory behaviour does not emerge and the wave behaves like a ray. This is the reason why for many years it was not understood if light was a wave or if it was made of particles.

[picture]

Figure 1.17. Diffraction pattern (reproduced from https://cronodon.com/Atomic/Photon.html).

1.6 Maxwell’s equations and electromagnetic waves

In this section, our purpose is to present Maxwell’s equations in integral form and to deduce from them their differential form [5, 6]. Since the mathematical tools needed for that are usually not included in basic calculus courses, we provide students with qualitative arguments only. In particular, we will apply our approach by referring to a simple arrangement of electric and magnetic fields, which will turn out to be useful to show that these fields must satisfy the d’Alembert equation.

1.6.1 The integral and the differential forms of Maxwell’s equations

Maxwell’s equations in integral form are expressed in terms of two fundamental integrated quantities associated with a given vector, that is, its line integral around a loop and its flux through a closed surface. The line integral is defined accordingly using the procedure illustrated in section 1.2 (see equation (1.4)). Concerning the flux Φ(A⃗) of a vector A⃗, for its definition one first divides a closed surface S into many elements of area ΔSi (the index i denotes the ith element), small enough that they can be considered planar and such that the value A⃗i of A⃗ can be assumed to be constant on each of them. Introducing the vector ΔS⃗i=ΔSnˆi, where the unit vector nˆi denotes the outward normal to ΔSi, one can define the sum

A⃗1·ΔS⃗1+A⃗2·ΔS⃗2+⋯+A⃗i·ΔA⃗i+⋯=∑iA⃗i·ΔS⃗i.

The flux of A⃗ through the closed surface S is then defined as the limit of this quantity when ΔSi→0:

Φ(A⃗)=limΔSi→0∑iA⃗i·ΔS⃗i=∯SF⃗·dS⃗.

The flux Φ(A⃗) is proportional to the density of the field lines and to the area through which it is calculated. In addition, it varies according to the direction of the flow with respect to the surface. We also note that the flux of a given vector keeps track of how many times that vector ‘pierces’ the surface, in the sense that a non-vanishing value of Φ signals the presence of sources or sinks inside the surface. In contrast, when the flux vanishes, for every field line entering the surface there is another one going out from it.

In terms of the above defined quantities, Maxwell’s equations in integral form are the following:

∯SE⃗·dS⃗=∑iQiintε0(1.29)

∮ΓE⃗·dl⃗=−dΦ(B⃗)dt(1.30)

∯SB⃗·dS⃗=0(1.31)

∮ΓB⃗·dl⃗=μ0∑jijint+ε0μ0dΦ(E⃗)dt.(1.32)

Equations (1.29) and (1.31) are the Gauss theorem for the electric and magnetic fields E⃗ and B⃗, respectively, with ∑iQiint being the total charge inside the closed surface S. Equation (1.30) is the Faraday–Neumann law relating the line integral of E⃗ around the loop Γ to the flux Φ of B⃗ through a surface having Γ as a boundary. Finally, equation (1.32) is the generalized Ampère theorem (also called the Ampère–Maxwell theorem), relating the line integral of B⃗ around the loop Γ to the algebraic sum ∑jijint of the electric currents passing through Γ and to the so-called displacement current expressed in terms of the flux of E⃗ enclosed by Γ. The constants ε0 and μ0 are the vacuum permittivity and the vacuum permeability, respectively.

Two fundamental theorems of mathematical analysis allow us to obtain Maxwell’s equations in differential form. They require the introduction of two quantities, known as the divergence of a vector and the curl of a vector, constructed as special combinations of the derivatives of the components of that vector. How can the related definitions be proposed to students? The first obstacle is to make them familiar with the differential operator nabla ∇⃗. For our purposes, it can be introduced simply as a mathematical tool defined as a strange vector without numerical coordinates but such that each component gives rise to the corresponding derivative when applied to a given function:

∇⃗=∂∂x,∂∂y,∂∂z.

In this way, if we have a vector A⃗=(Ax,Ay,Az), we can formally introduce the scalar product

∇⃗·A⃗=∂Ax∂x+∂Ay∂y+∂Az∂z,

which defines the divergence of A⃗, as well as the vectorial product

∇⃗×A⃗=Detiˆjˆkˆ∂∂x∂∂y∂∂zAxAyAz=∂Az∂y−∂Ay∂ziˆ+∂Ax∂z−∂Az∂xjˆ+∂Ay∂x−∂Ax∂ykˆ,

which defines the curl of A⃗.

In terms of these quantities, the two above-mentioned theorems, known as the divergence and the curl theorems, respectively, are expressed as

∯SA⃗·dS⃗=∫V(∇⃗·A⃗)dV(1.33)

∮ΓA⃗·dl⃗=∫S(∇⃗×A⃗)·dS⃗.(1.34)

The divergence theorem, equation (1.33), allows a surface integral defined on a closed surface S to be transformed into a volume integral over the volume V enclosed by S. Using the curl theorem, equation (1.34), a line integral over a closed curve Γ can be transformed into a surface integral defined on a generic surface S having Γ as a boundary.

We now provide arguments to show how a flux through a closed surface is linked to a divergence (theorem 1.33) and how a circulation is linked to a curl (theorem 1.34). This discussion closely follows that presented in [7]. Starting with the case of the divergence theorem, let us consider a small cube whose sides are given by Δx, Δy and Δz (Δx=Δy=Δz), as represented in figure 1.18. We want to find the flux of A⃗ through the surface of the cube summing the fluxes through each of the six faces. First consider the two faces, numbered 1 and 2 in the figure, perpendicular to the y-axis. Assuming that the cube is small enough that the value of A⃗ can be considered constant on each face, we have that the outward flux on the face 1 at y≠0 is Ay(1)ΔxΔz, while the outward flux at the opposite face is −Ay(2)ΔxΔz (in this case we have to consider the negative of the y-component of A⃗). The fact that the cube is very small also implies that the values of Ay(1) and Ay(2) are very close to each other, so that we can write

Ay(1)≃Ay(2)+∂Ay∂yΔy.

Summing the fluxes Φ(1) and Φ(2) through faces 1 and 2, respectively, we thus have

Φ(1)+Φ(2)=[Ay(1)−Ay(2)]ΔxΔz=∂Ay∂yΔxΔyΔz.

In the same way, we have for the flux through faces 3 and 4

Φ(3)+Φ(4)=∂Ay∂yΔxΔyΔz

and for the flux through faces 5 and 6

Φ(5)+Φ(6)=∂Ay∂yΔxΔyΔz.

We thus have that the total flux Φ through the six faces of the cube is

Φ=∂Ax∂x+∂Ay∂y+∂Az∂zΔxΔyΔz.(1.35)

This result tells us that the outward flux of a vector A⃗ from the surface of a very small cube is equal to the divergence of A⃗ multiplied by the volume of the cube. Going to a finite volume, we can use the fact that the total flux out of it is the sum of the fluxes out of each of the parts into which we imagine dividing that volume. This means that if we integrate equation (1.35) over the entire volume, we obtain the divergence theorem given by equation (1.33).

[picture]

Figure 1.18. A cube with sides Δx, Δy and Δz (Δx=Δy=Δz). The faces orthogonal to the axes x, y and z are numbered 5–6, 1–2 and 3–4, respectively.

Similarly, let us show how, according to equation (1.34), the flux of ∇⃗×A⃗ is connected to the circulation of A⃗. To this purpose, we evaluate the line integral of a given vector A⃗ around a small square such as the one in figure 1.19. In particular, we assume that A⃗ does not change much on each side of the square—of course, the smaller the square the better this assumption is. Moving from the lower left corner in the direction indicated by the arrow, we have that along the side marked 1 (see the figure), the line integral is

Γ1=A⃗(1)·Δx⃗=Ax(1)Δx.

Along the side marked 3 one has analogously

Γ3=A⃗·Δy⃗=−Ax(3)Δx,

where the minus sign takes into account that the line direction on side 3 is opposite to the one on side 1. Summing up these two contributions we have

Γ1+Γ3=[Ax(1)−Ax(3)]Δx.(1.36)

Since the square is very small, we can assume that the values Ax(1) and Ax(3) are very close. We can thus write

Ax(3)≃Ax(1)+∂Ay∂yΔy

so that equation (1.36) becomes

Γ1+Γ3=−∂Ax∂yΔxΔy.

Repeating the calculation for sides 2 and 4, one similarly has

Γ2+Γ4=∂Ay∂xΔxΔy

and thus the line integral Γ around the square takes the form

Γ=∑i=14Γi=∂Ay∂x−∂Ax∂yΔxΔy.(1.37)

We can see that the quantity in the parentheses is exactly the z-component of the curl of A⃗, while the product ΔxΔy is the area ΔS of the square. Considering that the z-component of A⃗ is also the component normal to the square, we have that equation (1.37) can be put in the form

Γ=∇⃗×A⃗·nˆΔS,(1.38)

where n⃗ˆ is the unit vector in the direction normal to the square. When we consider the circulation of a vector around any loop C, we can consider any surface having C as a boundary and divide it into a set of very small surface elements, small enough that they can be considered flat and very nearly a square. It is then possible to show that the sum of the circulations around the closed boundary of these very small squares equals the circulation around the original loop C, due to the cancellation of the contributions coming from the adjacent portions of the small squares. In this way, applying equation (1.38) to each of the small squares and summing up all the corresponding contributions, one recovers the curl theorem (1.34).

[picture]

Figure 1.19. A square with sides Δx and Δy (Δx=Δy) around which the circulation of a vector A⃗ is calculated. The sides parallel to the x-axis are numbered 1 and 3, while the sides parallel to the y-axis are numbered 2 and 4.

The application of the divergence theorem to the surface integrals appearing in Maxwell’s equations (1.29) and (1.31) allows these equations to be expressed in differential form. Introducing the charge density ρ associated with the charge in volume V,

∑iQiint=∫VρdV,

equation (1.29) becomes

∫V(∇⃗·E⃗)dV=1ε0∫VρdV

and thus, V being a generic volume,

∇⃗·E⃗=ρε0.

Similarly one obtains for B⃗

∇⃗·B⃗=0.

In the same way, the application of the curl theorem to the line integrals of E⃗ and B⃗ in equations (1.30) and (1.32), respectively, leads to

∇⃗×E⃗=−∂B⃗∂t(1.39)

∇⃗×B⃗=μ0j⃗+ε0μ0∂E⃗∂t.(1.40)

These two equations imply that variations in time of one of the two fields produce ‘rotational’ configurations of the other in planes which are orthogonal to the field which generates them.

1.6.2 Electromagnetic waves

We can now show that Maxwell’s equations establish that the dependence on time and space of the electric and the magnetic fields is wave-like [5]. Without loss of generality, we consider a simple case where the direction of propagation and the two fields E⃗ and B⃗ are mutually orthogonal:

According to this choice, equation (1.39) gives

∂E∂x=−∂B∂t.(1.42)

In the same way, from equation (1.40) one obtains

∂B∂x=−ε0μ0∂E∂t.(1.43)

Equations (1.41), (1.42) and (1.43) suggest that if one of the two fields varies in space the other varies in time, with the two fields remaining orthogonal to each other. The undulatory behaviour of the two fields can be verified by deriving equation (1.42) with respect to the spatial variable x,

∂2E∂x2=−∂∂x∂B∂t=−∂∂t∂B∂x,(1.44)

where one has taken into account that the order in which the derivatives with respect to different variables are performed is irrelevant. By substituting equation (1.43) in equation (1.44) we have

∂2E∂x2=ε0μ0∂2E∂t2.(1.45)

This is exactly the wave equation (1.25), with the constant factor ε0μ0 correctly having the dimension of the inverse of the square of a velocity. We can thus put

so that equation (1.45) takes the form

On the other hand, combining the two equations obtained by deriving equation (1.43) with respect to x and equation (1.42) with respect to t, we similarly obtain for the magnetic field

Equations (1.47) and (1.48) imply that the solutions E and B of Maxwell’s equations both satisfy the D’Alembert equation. It is thus confirmed that the two fields have wave-like behaviour. Together they constitute the so-called electromagnetic wave, propagating in a vacuum with a speed which does not depend on its frequency and wavelength. Its magnitude can be calculated by substituting the numerical values of the constants ε0 and μ0 in equation (1.46). We obtain

It is significant that the velocity of propagation of the electromagnetic waves is expressed in terms of the two fundamental constants, ε0 and μ0, appearing in the description of electric and magnetic phenomena, respectively. Its numerical value, as given in equation (1.49), is exactly equal to the speed of light in a vacuum, thus confirming that the light is an electromagnetic wave.

References

[1] Galilei G 2001 Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican (New York: Modern Library)

[2] Laplace P S 1995 A Philosophical Essay On Probabilities (New York: Dover)

[3] For a thorough presentation of the laws and applications of mechanics see, for example, Resnick R, Halliday D and Krane K S 2001 Physics vol 1 (New York: Wiley)

[4] Resnick R 1968 Introduction to Special Relativity (New York: Wiley)

[5] Halliday D, Resnick R and Krane K S 2001 Physics vol 2 (New York: Wiley)

[6] Jackson J D 1998 Classical Electrodynamics (New York: Wiley)

[7] Feynman R, Leighton R and Sands M 2011 The Feynman Lectures on Physics vol II (New York: Basic Books)