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

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:

• 1.  
  If two laboratories move with uniform linear motion, no experiment will give different results in either of them.
• 2.  
  A reference system moving with uniform linear motion is defined as inertial.
• 3.  
  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.

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:

• 1.  
  Do we need contact to exert a force?
• 2.  
  Does the application of a force move an object?
• 3.  
  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 $\vec{F}$ doing finite work) and its effect (the acceleration $\vec{a}$). 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 $\vec{F}=m\vec{a}$ 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 $\vec{F}=m\vec{a}$ along the vertical direction leads to
  $$\begin{eqnarray}&&{mg}-N-T={ma},\end{eqnarray} \tag{ 1.1 }$$
  where N is the normal force, T is the tension of the cable, a is the acceleration of the lift and $g=G\frac{{M}_{T}}{{r}^{2}}$ is the acceleration of gravity (G is the gravitational constant, M_T 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:
  $$\begin{eqnarray*}&&N={mg}-{ma}-T=m(g-a)-T.\end{eqnarray*}$$
  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 $\vec{F}=m\vec{a}$, because this force only affects the initial conditions of the motion, determining the value of the initial velocity $\mathit{\unicode[Book Antiqua]{x76}}$ ₀. More precisely, we deal in this case with an impulsive force acting during a very short time interval ${\rm{\Delta }}t$, which causes a variation of the quantity of motion in one dimension, given by
  $$\begin{eqnarray*}&&{\rm{\Delta }}p={{mv}}_{0}-0=F{\rm{\Delta }}t.\end{eqnarray*}$$
• Consider a pendulum and the projection of $\vec{F}=m\vec{a}$ in the direction of the cable; in some cases students are tempted to write
  $$\begin{eqnarray*}&&T-{mg}\,\cos \,\theta =0\end{eqnarray*}$$
  since there is no motion along the cable. Obviously, this is wrong since the normal component of the acceleration is clearly non-vanishing.

Zoom In Zoom Out Reset image size

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.

Zoom In Zoom Out Reset image size

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).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

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:

$$\begin{eqnarray}&&{m}_{1}{\vec{\mathit{\unicode[Book Antiqua]{x76}}}}_{1}+{m}_{2}{\vec{\mathit{\unicode[Book Antiqua]{x76}}}}_{2}=0\,\Rightarrow \,{m}_{1}{\vec{a}}_{1}=-{m}_{2}{\vec{a}}_{2}\,\Rightarrow \,{\vec{F}}_{1}=-{\vec{F}}_{2},\end{eqnarray} \tag{ 1.2 }$$

where ${\vec{F}}_{1}$ is the force that m₂ exerts on m₁ and ${\vec{F}}_{2}$ 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 ${\vec{F}}_{1}$ and ${\vec{F}}_{2}$ 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.

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:

$$\begin{eqnarray}&&{ \mathcal L }=\vec{F}\cdot {\rm{\Delta }}\vec{s}=F{\rm{\Delta }}s\,\cos \,\theta .\end{eqnarray} \tag{ 1.3 }$$

Here θ is the angle formed by the direction of the force $\vec{F}$ and the direction of the displacement ${\rm{\Delta }}\vec{s}$. 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 ${\rm{\Delta }}{\vec{s}}_{i}$ ($i=1,2,3,\ldots$) 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

$$\begin{eqnarray*}&&{\vec{F}}_{1}\cdot {\rm{\Delta }}{\vec{s}}_{1}+{\vec{F}}_{2}\cdot {\rm{\Delta }}{\vec{s}}_{2}+\cdots +{\vec{F}}_{i}\cdot {\rm{\Delta }}{\vec{s}}_{i}+\cdots =\sum _{i}{\vec{F}}_{i}\cdot {\rm{\Delta }}{\vec{s}}_{i}.\end{eqnarray*}$$

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:

$$\begin{eqnarray}&&{ \mathcal L }=\mathop{\mathrm{lim}}\limits_{}\,\sum _{i}{\vec{F}}_{i}\cdot {\rm{\Delta }}{\vec{s}}_{i}={\int }_{C}\vec{F}\cdot {\rm{d}}\vec{s}.\end{eqnarray} \tag{ 1.4 }$$

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 $\int$ is also used to indicate ordinary integrals, defined, according to figure 1.6, as

$$\begin{eqnarray}&&{\int }_{a}^{b}F\left(x\right){\rm{d}}x=\mathop{\mathrm{lim}}\limits_{{\rm{\Delta }}{x}_{i}\to 0}\,\sum _{i}F\left({\xi }_{i}\right){\rm{\Delta }}{x}_{i}.\end{eqnarray} \tag{ 1.5 }$$

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.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

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

$$\begin{eqnarray*}&&{ \mathcal L }={\int }_{{V}_{{\rm{A}}}}^{{V}_{{\rm{B}}}}P{\rm{d}}V,\end{eqnarray*}$$

where ${V}_{{\rm{A}}}$ and ${V}_{{\rm{B}}}$ are the values of V in the initial and final equilibrium states A and B, respectively. In the above expression, $P={F}_{\perp }/A$ 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.

Zoom In Zoom Out Reset image size

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:

$$\begin{eqnarray*}\begin{array}{rcl}{V}_{{\rm{B}}}-{V}_{{\rm{A}}} & = & -\,{\int }_{A}^{B}\vec{E}\cdot {\rm{d}}\vec{s}\\ \oint \vec{B}\cdot {\rm{d}}\vec{s} & = & {\mu }_{0}i.\end{array}\end{eqnarray*}$$

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

$$\begin{eqnarray}&&{\int }_{C}\vec{F}\cdot {\rm{d}}\vec{s}=V(f)-V(i),\end{eqnarray} \tag{ 1.6 }$$

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.

Kinetic energy is defined as the energy of an object of mass m in movement with speed equal to $\mathit{\unicode[Book Antiqua]{x76}}$ and it is expressed as $K=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}^{2}$. 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 $\vec{F}$ as the sum of all these forces, the second principle of dynamics asserts that $\vec{F}=m\vec{a}$, where $\vec{a}$ is the acceleration of the particle. Then, we can calculate the work ${ \mathcal L }$ done by $\vec{F}$ relative to a displacement from point A to point B:

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & {\int }_{{\rm{A}}}^{{\rm{B}}}\vec{F}\cdot {\rm{d}}\vec{s}={\int }_{{\rm{A}}}^{{\rm{B}}}m\vec{a}\cdot {\rm{d}}\vec{s}=m\,{\int }_{{\rm{A}}}^{{\rm{B}}}\frac{{\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}}{{\rm{d}}t}\cdot {\rm{d}}\vec{s}=m\,{\int }_{{\rm{A}}}^{{\rm{B}}}{\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot \vec{\mathit{\unicode[Book Antiqua]{x76}}}\\ & = & m{\int }_{{\rm{A}}}^{{\rm{B}}}\frac{{\rm{d}}({\mathit{\unicode[Book Antiqua]{x76}}}^{2})}{2}={\left.\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}^{2}\right|}_{{\rm{A}}}^{{\rm{B}}}=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{B}}}^{2}-\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{A}}}^{2}.\end{array}\end{eqnarray} \tag{ 1.7 }$$

Here, starting from $\vec{\mathit{\unicode[Book Antiqua]{x76}}}=\frac{{\rm{d}}\vec{s}}{{\rm{d}}t}$, we have used ${\rm{d}}({\mathit{\unicode[Book Antiqua]{x76}}}^{2})={\rm{d}}(\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot \vec{\mathit{\unicode[Book Antiqua]{x76}}})=\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot {\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}+{\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot \vec{\mathit{\unicode[Book Antiqua]{x76}}}$, so that ${\rm{d}}({\mathit{\unicode[Book Antiqua]{x76}}}^{2})=2\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot {\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}$. 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, $K=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}^{2}$, 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

$$\begin{eqnarray*}&&\vec{F}=\frac{{\rm{d}}\vec{p}}{{\rm{d}}t}=\frac{{\rm{d}}}{{\rm{d}}t}\frac{m\vec{\mathit{\unicode[Book Antiqua]{x76}}}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}}}\end{eqnarray*}$$

and calculating the derivative explicitly, we obtain

$$\begin{eqnarray*}&&\vec{F}=\frac{m}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}}}\frac{{\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}}{{\rm{d}}t}+\frac{m}{{c}^{2}}\frac{1}{{\left(1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}\right)}^{\frac{3}{2}}}\left(\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot \frac{{\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}}{{\rm{d}}t}\right)\vec{\mathit{\unicode[Book Antiqua]{x76}}}.\end{eqnarray*}$$

Given this expression of the relativistic force $\vec{F}$, we can evaluate the elementary work ${\rm{d}}{ \mathcal L }$. After some simple algebra, we obtain

$$\begin{eqnarray*}&&{\rm{d}}{ \mathcal L }=\vec{F}\cdot {\rm{d}}\vec{r}=\vec{F}\cdot \vec{\mathit{\unicode[Book Antiqua]{x76}}}{\rm{d}}t=\frac{m}{{\left(1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}\right)}^{\frac{3}{2}}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot {\rm{d}}\vec{\mathit{\unicode[Book Antiqua]{x76}}}.\end{eqnarray*}$$

This expression can also be written in the form

$$\begin{eqnarray}&&{\rm{d}}{ \mathcal L }={\rm{d}}\left(\frac{{{mc}}^{2}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}}}\right).\end{eqnarray} \tag{ 1.8 }$$

We can now evaluate the finite work ${ \mathcal L }$ done by $\vec{F}$ when a particle moves from A to B by integrating equation (1.8):

$$\begin{eqnarray*}&&{ \mathcal L }={\int }_{{\rm{A}}}^{{\rm{B}}}{\rm{d}}{ \mathcal L }={\int }_{{\rm{A}}}^{{\rm{B}}}{\rm{d}}\left(\frac{{{mc}}^{2}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}}}\right)=\frac{{{mc}}^{2}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{B}}}^{2}}{{c}^{2}}}}-\frac{{{mc}}^{2}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{A}}}^{2}}{{c}^{2}}}}.\end{eqnarray*}$$

This result allows the relativistic kinetic energy to be defined as

$$\begin{eqnarray*}&&K=\frac{{{mc}}^{2}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}}}+{K}_{0},\end{eqnarray*}$$

where K₀ is a constant. By imposing the condition $K(\mathit{\unicode[Book Antiqua]{x76}}=0)=0$ one immediately finds ${K}_{0}=-{{mc}}^{2}$, so that the final expression of the relativistic kinetic energy is

$$\begin{eqnarray*}&&K=\frac{{{mc}}^{2}}{\sqrt{1-\frac{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}{{c}^{2}}}}-{{mc}}^{2}.\end{eqnarray*}$$

It is worth noting that, expanding this expression in Taylor series for $\mathit{\unicode[Book Antiqua]{x76}}/c\ll 1$ and retaining terms up to second order, one recovers the classical expression $\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}^{2}$. 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 $\vec{F};$ for simplicity, we will consider a linear displacement ${\rm{\Delta }}\vec{s}$ from A to B and we will suppose that $\vec{F}$ remains constant along this linear path. Then, the work done by $\vec{F}$ is given by

$$\begin{eqnarray*}&&{ \mathcal L }=\vec{F}\cdot {\rm{\Delta }}\vec{s}=m\frac{{\rm{\Delta }}\vec{\mathit{\unicode[Book Antiqua]{x76}}}}{{\rm{\Delta }}t}\cdot {\rm{\Delta }}\vec{s}=m\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot {\rm{\Delta }}\vec{\mathit{\unicode[Book Antiqua]{x76}}}.\end{eqnarray*}$$

Since we are considering a linear displacement, vectors ${\rm{\Delta }}\vec{\mathit{\unicode[Book Antiqua]{x76}}}$ and $\vec{\mathit{\unicode[Book Antiqua]{x76}}}$ are parallel; then, $\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot {\rm{\Delta }}\vec{\mathit{\unicode[Book Antiqua]{x76}}}=\mathit{\unicode[Book Antiqua]{x76}}{\rm{\Delta }}\mathit{\unicode[Book Antiqua]{x76}}$ and we write

$$\begin{eqnarray*}&&{ \mathcal L }=m\vec{\mathit{\unicode[Book Antiqua]{x76}}}\cdot {\rm{\Delta }}\vec{\mathit{\unicode[Book Antiqua]{x76}}}=m\mathit{\unicode[Book Antiqua]{x76}}\,({\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{B}}}-{\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{A}}}).\end{eqnarray*}$$

In this equation, the speed $\mathit{\unicode[Book Antiqua]{x76}}$ is intended as the average speed $\frac{{\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{B}}}+{\mathit{\unicode[Book Antiqua]{x76}}}_{{\rm{A}}}}{2}$, then

$$\begin{eqnarray}&&{ \mathcal L }=m\,\frac{{\mathit{\unicode[Book Antiqua]{x76}}}_{B}+{\mathit{\unicode[Book Antiqua]{x76}}}_{A}}{2}\,({\mathit{\unicode[Book Antiqua]{x76}}}_{B}-{\mathit{\unicode[Book Antiqua]{x76}}}_{A})=\frac{1}{2}m\left({\mathit{\unicode[Book Antiqua]{x76}}}_{B}^{2}-{\mathit{\unicode[Book Antiqua]{x76}}}_{A}^{2}\right)=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{B}^{2}-\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{A}^{2}.\end{eqnarray} \tag{ 1.9 }$$

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.

Zoom In Zoom Out Reset image size

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

$$\begin{eqnarray}&&{ \mathcal L }=-[U(B)-U(A)].\end{eqnarray} \tag{ 1.10 }$$

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 ${ \mathcal L }$ the work ${{ \mathcal L }}_{\mathrm{cons}}$ done by conservative forces from the work ${{ \mathcal L }}_{\mathrm{non}\unicode{x02010}\mathrm{cons}}$ done by non-conservative forces:

$$\begin{eqnarray*}&&{{ \mathcal L }}_{\mathrm{cons}}+{{ \mathcal L }}_{\mathrm{non}\unicode{x02010}\mathrm{cons}}=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{B}^{2}-\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{A}^{2}.\end{eqnarray*}$$

Then, using equation (1.10) one has

$$\begin{eqnarray*}&&-[U\text{(B)}-U\text{(A)}]+{{ \mathcal L }}_{\text{non}\unicode{x02010}\text{cons}}=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{B}^{2}-\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{A}^{2}.\end{eqnarray*}$$

Rewriting this equation in the form

$$\begin{eqnarray}&&{{ \mathcal L }}_{\text{non}\unicode{x02010}\text{cons}}=\left[\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{B}^{2}+U\text{(B)}\right]-\left[\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{A}^{2}+U\text{(A)}\right],\end{eqnarray} \tag{ 1.11 }$$

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

$$\begin{eqnarray}&&E(P)=\frac{1}{2}{m\mathit{\unicode[Book Antiqua]{x76}}}_{P}^{2}+U(P),\end{eqnarray} \tag{ 1.12 }$$

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.

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.

Zoom In Zoom Out Reset image size

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

$$\begin{eqnarray}&&\alpha =f(x\pm \mathit{\unicode[Book Antiqua]{x76}}t).\end{eqnarray} \tag{ 1.17 }$$

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 $x\pm \mathit{\unicode[Book Antiqua]{x76}}t$, where $\mathit{\unicode[Book Antiqua]{x76}}$ 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 $f(x-\mathit{\unicode[Book Antiqua]{x76}}t)$ and $f(x+\mathit{\unicode[Book Antiqua]{x76}}t)$, 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:

$$\begin{eqnarray}&&y(x,t)=A\,\sin \,k(x-\mathit{\unicode[Book Antiqua]{x76}}t).\end{eqnarray} \tag{ 1.18 }$$

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

$$\begin{eqnarray}&&y(x,t)=A\,\sin ({kx}-k\mathit{\unicode[Book Antiqua]{x76}}t),\end{eqnarray} \tag{ 1.19 }$$

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

$$\begin{eqnarray}&&\lambda =\frac{2\pi }{k},\end{eqnarray} \tag{ 1.20 }$$

called the wavelength, represents the spatial period of the wave. As a consequence, $k=\frac{2\pi }{\lambda }$ 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.

Zoom In Zoom Out Reset image size

On the other hand, $k\mathit{\unicode[Book Antiqua]{x76}}$ is linked to the temporal periodicity of the wave. For a fixed value of x, one has that when $k\mathit{\unicode[Book Antiqua]{x76}}t=2\pi n$, n being again an integer, the function (1.19) repeats itself in time. Then, waiting for a time equal to an integer multiple of

$$\begin{eqnarray}&&T=\frac{2\pi }{k\mathit{\unicode[Book Antiqua]{x76}}},\end{eqnarray} \tag{ 1.21 }$$

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

$$\begin{eqnarray}&&\nu =\frac{1}{T}=\frac{k\mathit{\unicode[Book Antiqua]{x76}}}{2\pi }.\end{eqnarray} \tag{ 1.22 }$$

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

$$\begin{eqnarray*}&&\omega =2\pi \nu .\end{eqnarray*}$$

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

$$\begin{eqnarray*}&&\omega =2\pi \frac{1}{T}=k\mathit{\unicode[Book Antiqua]{x76}}=\frac{2\pi }{\lambda }\mathit{\unicode[Book Antiqua]{x76}}.\end{eqnarray*}$$

From the previous equations, we also obtain

$$\begin{eqnarray*}&&\mathit{\unicode[Book Antiqua]{x76}}=\frac{\lambda }{T}.\end{eqnarray*}$$

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:

• 1.  
  A wave is always described in terms of a function of space and time variables appearing in the special combination $(x\pm \mathit{\unicode[Book Antiqua]{x76}}t).$
• 2.  
  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 $\mathit{\unicode[Book Antiqua]{x76}}t$ (point 1) on an equal footing and it has to be linear (point 2).

Zoom In Zoom Out Reset image size

We propose a heuristic approach to derive this equation, limiting ourselves for simplicity to waves propagating in one dimension with speed equal to $\mathit{\unicode[Book Antiqua]{x76}}$. 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 $\mathit{\unicode[Book Antiqua]{x76}}t$ does. A starting point may be

$$\begin{eqnarray}&&\frac{{\rm{\Delta }}f}{{\rm{\Delta }}x}=\frac{{\rm{\Delta }}f}{{\rm{\Delta }}(\mathit{\unicode[Book Antiqua]{x76}}t)}.\end{eqnarray} \tag{ 1.23 }$$

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

$$\begin{eqnarray}&&\frac{\partial f}{\partial x}=\frac{1}{\mathit{\unicode[Book Antiqua]{x76}}}\frac{\partial f}{\partial t}.\end{eqnarray} \tag{ 1.24 }$$

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 $+\mathit{\unicode[Book Antiqua]{x76}}t$ and $-\mathit{\unicode[Book Antiqua]{x76}}t$ 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:

$$\begin{eqnarray}&&\frac{{\partial }^{2}f}{\partial {x}^{2}}=\frac{1}{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}\frac{{\partial }^{2}f}{\partial {t}^{2}}.\end{eqnarray} \tag{ 1.25 }$$

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.

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

$$\begin{eqnarray}&&{y}_{1}(x,t)=A\,\sin ({kx}-\omega t+\varphi )\end{eqnarray} \tag{ 1.26 }$$

$$\begin{eqnarray}&&{y}_{2}(x,t)=A\,\sin ({kx}-\omega t),\end{eqnarray} \tag{ 1.27 }$$

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

$$\begin{eqnarray}&&y(x,t)={y}_{1}(x,t)+{y}_{2}(x,t)=2A\,\cos \,\frac{\varphi }{2}\,\sin \,\left({kx}-\omega t+\frac{\varphi }{2}\right),\end{eqnarray} \tag{ 1.28 }$$

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),

$$\begin{eqnarray*}&&{A}_{0}=2A\,\cos \,\frac{\varphi }{2},\end{eqnarray*}$$

is equal to the sum of the amplitudes of equations (1.26) and (1.27) only in the particular case in which $\varphi =0$ or, in general, when $\varphi =2n\pi$, n being an integer. This is the case of constructive interference. In contrast, if $\varphi =\pi /2$ or, more generally, $\varphi =2\pi (n+\frac{1}{2})$, 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.

Zoom In Zoom Out Reset image size

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 $\lambda \sim d$, where λ is the wavelength of the wave and d represents the dimension of the slit. Conversely, when $\lambda \gg d$, the wave moves undisturbed, as if the obstacle was not there, whereas if $\lambda \ll d$, 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.

Zoom In Zoom Out Reset image size

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 ${\rm{\Phi }}(\vec{A})$ of a vector $\vec{A}$, for its definition one first divides a closed surface S into many elements of area ${\rm{\Delta }}{S}_{i}$ (the index i denotes the ith element), small enough that they can be considered planar and such that the value ${\vec{A}}_{i}$ of $\vec{A}$ can be assumed to be constant on each of them. Introducing the vector ${\rm{\Delta }}{\vec{S}}_{i}={\rm{\Delta }}S\,{\hat{n}}_{i}$, where the unit vector ${\hat{n}}_{i}$ denotes the outward normal to ${\rm{\Delta }}{S}_{i}$, one can define the sum

$$\begin{eqnarray*}&&{\vec{A}}_{1}\cdot {\rm{\Delta }}{\vec{S}}_{1}+{\vec{A}}_{2}\cdot {\rm{\Delta }}{\vec{S}}_{2}+\cdots +{\vec{A}}_{i}\cdot {\rm{\Delta }}{\vec{A}}_{i}+\cdots =\sum _{i}{\vec{A}}_{i}\cdot {\rm{\Delta }}{\vec{S}}_{i}.\end{eqnarray*}$$

The flux of $\vec{A}$ through the closed surface S is then defined as the limit of this quantity when ${\rm{\Delta }}{S}_{i}\to 0$:

$$\begin{eqnarray*}&&{\rm{\Phi }}(\vec{A})=\mathop{\mathrm{lim}}\limits_{{\rm{\Delta }}{S}_{i}\to 0}\,\sum _{i}{\vec{A}}_{i}\cdot {\rm{\Delta }}{\vec{S}}_{i}={\unicode{x0222F} }_{S}\vec{F}\cdot {\rm{d}}\vec{S}.\end{eqnarray*}$$

The flux ${\rm{\Phi }}(\vec{A})$ 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:

$$\begin{eqnarray}&&{\unicode{x0222F} }_{S}\,\vec{E}\cdot {\rm{d}}\vec{S}=\frac{\sum _{i}{Q}_{i}^{\mathrm{int}}}{{\varepsilon }_{0}}\end{eqnarray} \tag{ 1.29 }$$

$$\begin{eqnarray}&&{\oint }_{{\rm{\Gamma }}}\,\vec{E}\cdot {\rm{d}}\vec{l}=-\frac{\mathrm{d\Phi }(\vec{B})}{{\rm{d}}t}\end{eqnarray} \tag{ 1.30 }$$

$$\begin{eqnarray}&&{\unicode{x0222F} }_{S}\,\vec{B}\cdot {\rm{d}}\vec{S}=0\end{eqnarray} \tag{ 1.31 }$$

$$\begin{eqnarray}&&{\oint }_{{\rm{\Gamma }}}\,\vec{B}\cdot {\rm{d}}\vec{l}={\mu }_{0}\sum _{j}{i}_{j}^{\mathrm{int}}+{\varepsilon }_{0}{\mu }_{0}\frac{\mathrm{d\Phi }(\vec{E})}{{\rm{d}}t}.\end{eqnarray} \tag{ 1.32 }$$

Equations (1.29) and (1.31) are the Gauss theorem for the electric and magnetic fields $\vec{E}$ and $\vec{B}$, respectively, with ${\sum }_{i}{Q}_{i}^{\mathrm{int}}$ being the total charge inside the closed surface S. Equation (1.30) is the Faraday–Neumann law relating the line integral of $\vec{E}$ around the loop Γ to the flux Φ of $\vec{B}$ 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 $\vec{B}$ around the loop Γ to the algebraic sum ${\sum }_{j}{i}_{j}^{\mathrm{int}}$ of the electric currents passing through Γ and to the so-called displacement current expressed in terms of the flux of $\vec{E}$ enclosed by Γ. The constants ${\varepsilon }_{0}$ and ${\mu }_{0}$ 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 $\vec{{\rm{\nabla }}}$. 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:

$$\begin{eqnarray*}&&\vec{{\rm{\nabla }}}=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right).\end{eqnarray*}$$

In this way, if we have a vector $\vec{A}=({A}_{x},{A}_{y},{A}_{z})$, we can formally introduce the scalar product

$$\begin{eqnarray*}&&\vec{{\rm{\nabla }}}\cdot \vec{A}=\frac{\partial {A}_{x}}{\partial x}+\frac{\partial {A}_{y}}{\partial y}+\frac{\partial {A}_{z}}{\partial z},\end{eqnarray*}$$

which defines the divergence of $\vec{A}$, as well as the vectorial product

$$\begin{eqnarray*}\begin{array}{rcl}\vec{{\rm{\nabla }}}\times \vec{A} & = & \mathrm{Det}\,\left(\begin{array}{rcl}\hat{i} & \hat{j} & \hat{k}\\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ {A}_{x} & {A}_{y} & {A}_{z}\end{array}\right)\\ & = & \left(\frac{\partial {A}_{z}}{\partial y}-\frac{\partial {A}_{y}}{\partial z}\right)\hat{i}+\left(\frac{\partial {A}_{x}}{\partial z}-\frac{\partial {A}_{z}}{\partial x}\right)\hat{j}+\left(\frac{\partial {A}_{y}}{\partial x}-\frac{\partial {A}_{x}}{\partial y}\right)\hat{k},\end{array}\end{eqnarray*}$$

which defines the curl of $\vec{A}$.

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

$$\begin{eqnarray}&&{\unicode{x0222F} }_{S}\vec{A}\cdot {\rm{d}}\vec{S}={\int }_{V}(\vec{{\rm{\nabla }}}\cdot \vec{A}){\rm{d}}V\end{eqnarray} \tag{ 1.33 }$$

$$\begin{eqnarray}&&{\oint }_{{\rm{\Gamma }}}\vec{A}\cdot {\rm{d}}\vec{l}={\int }_{S}(\vec{{\rm{\nabla }}}\times \vec{A})\cdot {\rm{d}}\vec{S}.\end{eqnarray} \tag{ 1.34 }$$

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 ${\rm{\Delta }}x$, ${\rm{\Delta }}y$ and ${\rm{\Delta }}z$ (${\rm{\Delta }}x={\rm{\Delta }}y={\rm{\Delta }}z$), as represented in figure 1.18. We want to find the flux of $\vec{A}$ 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 $\vec{A}$ can be considered constant on each face, we have that the outward flux on the face 1 at $y\ne 0$ is ${A}_{y}(1){\rm{\Delta }}x{\rm{\Delta }}z$, while the outward flux at the opposite face is $-{A}_{y}(2){\rm{\Delta }}x{\rm{\Delta }}z$ (in this case we have to consider the negative of the y-component of $\vec{A}$). The fact that the cube is very small also implies that the values of ${A}_{y}(1)$ and ${A}_{y}(2)$ are very close to each other, so that we can write

$$\begin{eqnarray*}&&{A}_{y}(1)\simeq {A}_{y}(2)+\frac{\partial {A}_{y}}{\partial y}\,{\rm{\Delta }}y.\end{eqnarray*}$$

Summing the fluxes ${\rm{\Phi }}(1)$ and ${\rm{\Phi }}(2)$ through faces 1 and 2, respectively, we thus have

$$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Phi }}(1)+{\rm{\Phi }}(2) & = & [{A}_{y}(1)-{A}_{y}(2)]{\rm{\Delta }}x{\rm{\Delta }}z\\ & = & \frac{\partial {A}_{y}}{\partial y}{\rm{\Delta }}x{\rm{\Delta }}y{\rm{\Delta }}z.\end{array}\end{eqnarray*}$$

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

$$\begin{eqnarray*}&&{\rm{\Phi }}(3)+{\rm{\Phi }}(4)=\frac{\partial {A}_{y}}{\partial y}{\rm{\Delta }}x{\rm{\Delta }}y{\rm{\Delta }}z\end{eqnarray*}$$

and for the flux through faces 5 and 6

$$\begin{eqnarray*}&&{\rm{\Phi }}(5)+{\rm{\Phi }}(6)=\frac{\partial {A}_{y}}{\partial y}{\rm{\Delta }}x{\rm{\Delta }}y{\rm{\Delta }}z.\end{eqnarray*}$$

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

$$\begin{eqnarray}&&{\rm{\Phi }}=\left(\frac{\partial {A}_{x}}{\partial x}+\frac{\partial {A}_{y}}{\partial y}+\frac{\partial {A}_{z}}{\partial z}\right){\rm{\Delta }}x{\rm{\Delta }}y{\rm{\Delta }}z.\end{eqnarray} \tag{ 1.35 }$$

This result tells us that the outward flux of a vector $\vec{A}$ from the surface of a very small cube is equal to the divergence of $\vec{A}$ 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).

Zoom In Zoom Out Reset image size

Similarly, let us show how, according to equation (1.34), the flux of $\vec{{\rm{\nabla }}}\times \vec{A}$ is connected to the circulation of $\vec{A}$. To this purpose, we evaluate the line integral of a given vector $\vec{A}$ around a small square such as the one in figure 1.19. In particular, we assume that $\vec{A}$ 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

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{1}=\vec{A}(1)\cdot {\rm{\Delta }}\vec{x}={A}_{x}(1){\rm{\Delta }}x.\end{eqnarray*}$$

Along the side marked 3 one has analogously

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{3}=\vec{A}\cdot {\rm{\Delta }}\vec{y}=-{A}_{x}(3){\rm{\Delta }}x,\end{eqnarray*}$$

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

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{3}=[{A}_{x}(1)-{A}_{x}(3)]{\rm{\Delta }}x.\end{eqnarray} \tag{ 1.36 }$$

Since the square is very small, we can assume that the values ${A}_{x}(1)$ and ${A}_{x}(3)$ are very close. We can thus write

$$\begin{eqnarray*}&&{A}_{x}(3)\simeq {A}_{x}(1)+\frac{\partial {A}_{y}}{\partial y}{\rm{\Delta }}y\end{eqnarray*}$$

so that equation (1.36) becomes

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{3}=-\frac{\partial {A}_{x}}{\partial y}{\rm{\Delta }}x{\rm{\Delta }}y.\end{eqnarray*}$$

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

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{2}+{{\rm{\Gamma }}}_{4}=\frac{\partial {A}_{y}}{\partial x}{\rm{\Delta }}x{\rm{\Delta }}y\end{eqnarray*}$$

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

$$\begin{eqnarray}&&{\rm{\Gamma }}=\sum _{i=1}^{4}{{\rm{\Gamma }}}_{i}=\left(\frac{\partial {A}_{y}}{\partial x}-\frac{\partial {A}_{x}}{\partial y}\right){\rm{\Delta }}x{\rm{\Delta }}y.\end{eqnarray} \tag{ 1.37 }$$

We can see that the quantity in the parentheses is exactly the z-component of the curl of $\vec{A}$, while the product ${\rm{\Delta }}x{\rm{\Delta }}y$ is the area ${\rm{\Delta }}S$ of the square. Considering that the z-component of $\vec{A}$ is also the component normal to the square, we have that equation (1.37) can be put in the form

$$\begin{eqnarray}&&{\rm{\Gamma }}=\left(\vec{{\rm{\nabla }}}\times \vec{A}\right)\cdot \hat{n}{\rm{\Delta }}S,\end{eqnarray} \tag{ 1.38 }$$

where $\hat{\vec{n}}$ 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).

Zoom In Zoom Out Reset image size

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,

$$\begin{eqnarray*}&&\sum _{i}{Q}_{i}^{\mathrm{int}}={\int }_{V}\rho {\rm{d}}V,\end{eqnarray*}$$

equation (1.29) becomes

$$\begin{eqnarray*}&&{\int }_{V}(\vec{{\rm{\nabla }}}\cdot \vec{E}){\rm{d}}V=\frac{1}{{\varepsilon }_{0}}{\int }_{V}\rho {\rm{d}}V\end{eqnarray*}$$

and thus, V being a generic volume,

$$\begin{eqnarray*}&&\vec{{\rm{\nabla }}}\cdot \vec{E}=\frac{\rho }{{\varepsilon }_{0}}.\end{eqnarray*}$$

Similarly one obtains for $\vec{B}$

$$\begin{eqnarray*}&&\vec{{\rm{\nabla }}}\cdot \vec{B}=0.\end{eqnarray*}$$

In the same way, the application of the curl theorem to the line integrals of $\vec{E}$ and $\vec{B}$ in equations (1.30) and (1.32), respectively, leads to

$$\begin{eqnarray}&&\vec{{\rm{\nabla }}}\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\end{eqnarray} \tag{ 1.39 }$$

$$\begin{eqnarray}&&\vec{{\rm{\nabla }}}\times \vec{B}={\mu }_{0}\vec{j}+{\varepsilon }_{0}{\mu }_{0}\frac{\partial \vec{E}}{\partial t}.\end{eqnarray} \tag{ 1.40 }$$

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.

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 $\vec{E}$ and $\vec{B}$ are mutually orthogonal:

$$\begin{eqnarray}&&\vec{\mathit{\unicode[Book Antiqua]{x76}}}=\mathit{\unicode[Book Antiqua]{x76}}\hat{i}\qquad \vec{E}=E\hat{j}\qquad \vec{B}=B\hat{k}.\end{eqnarray} \tag{ 1.41 }$$

According to this choice, equation (1.39) gives

$$\begin{eqnarray}&&\frac{\partial E}{\partial x}=-\frac{\partial B}{\partial t}.\end{eqnarray} \tag{ 1.42 }$$

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

$$\begin{eqnarray}&&\frac{\partial B}{\partial x}=-{\varepsilon }_{0}{\mu }_{0}\frac{\partial E}{\partial t}.\end{eqnarray} \tag{ 1.43 }$$

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,

$$\begin{eqnarray}&&\frac{{\partial }^{2}E}{\partial {x}^{2}}=-\frac{\partial }{\partial x}\frac{\partial B}{\partial t}=-\frac{\partial }{\partial t}\frac{\partial B}{\partial x},\end{eqnarray} \tag{ 1.44 }$$

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

$$\begin{eqnarray}&&\frac{{\partial }^{2}E}{\partial {x}^{2}}={\varepsilon }_{0}{\mu }_{0}\frac{{\partial }^{2}E}{\partial {t}^{2}}.\end{eqnarray} \tag{ 1.45 }$$

This is exactly the wave equation (1.25), with the constant factor ${\varepsilon }_{0}{\mu }_{0}$ correctly having the dimension of the inverse of the square of a velocity. We can thus put

$$\begin{eqnarray}&&\mathit{\unicode[Book Antiqua]{x76}}=\frac{1}{\sqrt{{\varepsilon }_{0}{\mu }_{0}}}\end{eqnarray} \tag{ 1.46 }$$

so that equation (1.45) takes the form

$$\begin{eqnarray}&&\frac{{\partial }^{2}E}{\partial {x}^{2}}=\frac{1}{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}\frac{{\partial }^{2}E}{\partial {t}^{2}}.\end{eqnarray} \tag{ 1.47 }$$

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

$$\begin{eqnarray}&&\frac{{\partial }^{2}B}{\partial {x}^{2}}=\frac{1}{{\mathit{\unicode[Book Antiqua]{x76}}}^{2}}\frac{{\partial }^{2}B}{\partial {t}^{2}}.\end{eqnarray} \tag{ 1.48 }$$

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 ${\varepsilon }_{0}$ and ${\mu }_{0}$ in equation (1.46). We obtain

$$\begin{eqnarray}&&\mathit{\unicode[Book Antiqua]{x76}}=\frac{1}{\sqrt{\left(8.85\times {10}^{-12}\frac{{\rm{C}}}{{\rm{N}}\,{{\rm{m}}}^{2}}\right)\left(4\pi \times {10}^{-7}\frac{{\rm{T}}\,{\rm{m}}}{{\rm{A}}}\right)}}\cong 3\times {10}^{8}\,{\mathrm{ms}}^{-1}.\end{eqnarray} \tag{ 1.49 }$$

It is significant that the velocity of propagation of the electromagnetic waves is expressed in terms of the two fundamental constants, ${\varepsilon }_{0}$ and ${\mu }_{0}$, 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.