Section 12–4 Fundamental forces. Fields

(Electrical force / Gravitational force / Magnetic force)

In this section, Feynman discusses electrical force, gravitational force, and magnetic force. These forces are fundamental in the sense that their laws are fundamentally simple. Physics teachers may clarify that there are four fundamental forces: strong, weak, electromagnetic, and gravitational.

1. Electrical force:

“…if the magnitudes of the charges are q₁ and q₂, respectively, the force varies inversely as the square of the distance between the charges, or F = (const)q₁q₂/r² (Feynman et al., 1963, section 12–4 Fundamental forces. Fields).”

Coulomb’s law of electrostatics states that the magnitude of electrostatic force between two stationary point charges in a vacuum that possess q₁ and q₂ varies inversely as the square of the distance between the two charges, F = q₁q₂/4pϵ₀r², in which ϵ₀ = 8.854×10⁻¹² C²/Nm². For unlike charges, this law is similar to Newton’s law of gravitation, but for like charges the force is repulsive and the direction is reversed. An important idealization of Coulomb’s law is that the interacting charged objects are stationary. In addition, approximation methods are needed to calculate the complex forces because the charged objects move in a complicated way.

The net electric field of a number of sources is equal to the vector sum of the electric field of the individual source in accordance with the superposition principle. Apparent limitations of this principle can be shown by analysis that it is due to the oversight of certain moving charges. Using the field concept, Feynman explains that the charge q₁ at P creates a “condition” at R, such that when the charge q₂ is placed at R it “feels” an electrical force. In short, a charge q₁ affects the space around another charge q₂ and cause how the charge q₂ moves. The so-called empty space “is an explosive environment, ready to burst forth with real quark-antiquark molecules (Wilczek, 2008, p. 91).” This is in contrast to a slogan that Feynman had, “[t]he vacuum is empty. It weighs nothing because there's nothing there (Wilczek, 1999, p. 13).”

2. Gravitational force:

“In the case of gravitation, we can do exactly the same thing. In this case, where the force F = −Gm₁m₂r/r³, we can make an analogous analysis, as follows: the force on a body in a gravitational field is the mass of that body times the field C (Feynman et al., 1963, section 12–4 Fundamental forces. Fields).”

Similarly, we can analyze the gravitational force, F = −Gm₁m₂r/r³, as follows: the gravitational force on a body near a massive object is the mass of the body times the gravitational field C due to the massive object. Essentially, the gravitational force on a body of mass m₂ is equal to m₂ times the field C produced by m₁, F = m₂C. The gravitational field C produced by a body of mass m₁ is C = −Gm₁r/r³ and it is directed radially similar to the electrical forces. An important idealization of Newton’s law of gravitation is that the bodies are stationary. In general, an approximation method is to assume astronomical bodies are perfectly spherical. Furthermore, Feynman mentions that the superposition principle is not exactly applicable to gravity because Newton’s law of gravitation is approximately correct.

By using the field concept, Feynman says that a body that has mass m₁ creates a field C in all the surrounding space, such that the force on m₂ is given by F = m₂C. To be precise, Newton’s law of gravitation has its limitations and it is superseded by Einstein’s law of gravitation. In a lecture on quantum gravity, Feynman asks, “What if Einstein had never been born, and the General Theory of Relativity had never been created? How would high energy theorists understand gravity?” (Zweig, 2018) A more important question is whether the gravitational force between moving objects (more energy content) is larger or smaller than static objects? Feynman explains that “the spin-0 theory is out, and we need spin 2 in order to have a theory in which the attraction will be proportional to the energy content (Feynman et al., 1995, p. 31).”

3. Magnetic force:

“Closely related to electrical force is another kind, called magnetic force, and this too is analyzed in terms of a field (Feynman et al., 1963, section 12–4 Fundamental forces. Fields).”

Relations between electrical and magnetic forces can be illustrated using an electron-ray tube experiment. If the components of the electric field E and the magnetic induction B are (E_x, E_y, E_z) and (B_x, B_y, B_z) respectively, and if the velocity v has the components (v_x, v_y, v_z), then the Lorentz force law for a moving charge q has the components, F_x = q(E_x + v_yB_z – v_zB_y), F_y = q(E_y + v_zB_x − v_xB_z), F_z = q(E_z + v_xB_y − v_yB_x). An idealization of magnetic force is that a long wire is imagined to be an infinitely long wire. Next, physicists need approximation methods such as assuming the velocity of a moving particle is constant and the magnetic field strength is uniform within a small region of space. Notably, the equation, F_B = qv X B, is not violated for particles that move close to the speed of light.

Griffiths (1999) opines that the term magnetic induction is an absurd choice because it has at least two other meanings in electrodynamics. Magnetic induction is also known as magnetic flux density (B) that is different from magnetic field strength (H). (To avoid confusions or argument, one may simply write H-field and B-field.) Although B-field is commonly defined in terms of the equation, F_B = qv X B, some students may feel uncomfortable with the cross product; note that B is actually a pseudo-vector. Thus, one suggestion is to define B using the equation B = F/q_m, whereby q_m is the unit test magnetic pole (Wellner, 1992). However, it is debatable whether the magnetic field should be defined in terms of magnetic monopoles.

Questions for discussion:

1. How would you define an electrical force and electrical field?

2. How would you define a gravitational force and gravitational field?

3. How would you define a magnetic force and magnetic field?

The moral of the lesson: there are idealizations, approximations, and limitations in defining an electric force, gravitational force, and magnetic force.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

3. Griffiths, D. J. (1999). Introduction to Electrodynamics (3rd Edition). Upper Saddle River, N. J.: Prentice Hall.

4. Wellner, M. (1992). Reflections on V×B. American Journal of Physics, 60(9), 777.

5. Wilczek, F. (1999). The persistence of ether. Physics Today, 52(1), 11-13.

6. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.

7. Zweig, G. (2018). Remembering Feynman. Retrieved from

https://www.marinabaysands.com/museum/richard-feynman/essays.html

Section 12–3 Molecular forces

(Attractive force / Repulsive force / Proportional to the displacement)

In this section, Feynman discusses how molecular forces become attractive and repulsive as well as how molecular forces are proportional to the displacement.

1. Attractive force:

“For all nonpolar molecules, in which all the electrical forces are neutralized, it nevertheless turns out that the force at very large distances is an attraction and varies inversely as the seventh power of the distance… (Feynman et al., 1963, section 12–3 Molecular forces).”

Feynman states that molecular forces are forces between the atoms and are the origin of frictional forces. In a water molecule, the negative charges are located nearer to the oxygen, but the mean positions of the negative charges and of the positive charges are not at the same point; thus, a molecule nearby feels a dipole-dipole force. The molecular force at large distances is attractive and varies inversely as the seventh power of the distance, F = -k/r⁷, where k is a constant depending on the molecules. The molecular forces (or Van der Waals forces) can be classified as three types: (1) Keesom interaction (permanent dipole-permanent dipole interaction), (2) Debye interaction (permanent dipole-induced dipole interaction), and (3) London interaction (induced dipole-induced dipole interaction). Molecular forces can be demonstrated by a friction experiment using a sliding glass tumbler or Johansson blocks.

When Feynman was about twenty-one years old, he published his undergraduate thesis at MIT in The Physical Review. In his own words, “Van der Waals’ forces can also be interpreted as arising from charge distributions with higher concentration between the nuclei. The Schrodinger perturbation theory for two interacting atoms at a separation R, large compared to the radii of the atoms, leads to the result that the charge distribution of each is distorted from central symmetry, a dipole moment of order 1/R⁷ being induced in each atom. The negative charge distribution of each atom has its center of gravity moved slightly toward the other. It is not the interaction of these dipoles which leads to van der Waals' force, but rather the attraction of each nucleus for the distorted charge distribution of its own electrons that gives the attractive 1/R⁷ force… (Feynman, 1939, p. 343).” In 1936, Hans Hellman derived a molecular force theorem and it is now known as Hellman-Feynman theorem.

2. Repulsive force:

“When atoms or molecules get too close they repel with a very large repulsion; that is what keeps us from falling through the floor! (Feynman et al., 1963, section 12–3 Molecular forces).”

By using a graph as shown in Fig. 12–2, it illustrates that the molecular forces attract at long distances and repel at short distances. This indicates that all the atoms are held together by their attractions to form solids, but held apart by their repulsions that set in when they are too close together. More important, when atoms or molecules are closer together, they repel with a very large repulsion. This explains what keeps us from falling through the floor. Feynman elaborates that at a certain distance d, the forces are zero where the graph crosses the axis. It means that the molecular forces are all balanced such that the molecules stay at that distance apart from one another. However, Feynman did not include the r⁻¹² term, which represents the repulsive molecular force.

Some teachers explain that the r⁻¹² term, which is the repulsive term, is due to Pauli repulsion at short ranges as a result of overlapping electron orbitals. In Statistical Mechanics, Huang (1987) writes that “[t]he attractive part of the potential energy originates from the mutual electric polarization of the two molecules and the repulsive part from the Coulomb repulsion of the overlapping electronic clouds of the molecules (p. 38).” Feynman did not use Pauli exclusion principle to elaborate the repulsive forces possibly because he was unable to find a simple way of explaining Pauli’s principle at an elementary level. In his words, “[t]his probably means that we do not have a complete understanding of the fundamental principle involved (Feynman et al., 1966, Section 4–1 Bose particles and Fermi particles).”

3. Proportional to the displacement:

“Therefore, in many circumstances, if the displacement is not too great the force is proportional to the displacement (Feynman et al., 1963, section 12–3 Molecular forces).”

If molecules are separated by a very short distance closer or farther than their equilibrium positions, then the force is proportional to the displacement. This principle is known as Hooke’s law of elasticity, which specifies that the restoring force of a body is proportional to the extension of the body. The validity of Hooke’s law is dependent on the materials; for instance, the force on dough or putty is quite small, but the force on a piece of steel is relatively large. Feynman explains that Hooke’s law can be experimentally demonstrated with a vertically suspended coil spring that is long and made of steel. An advantage of using a long coil spring allows us to measure the extension of the spring that is relatively short. Furthermore, the length of the spring that is made of steel may increase the total force to extend the spring due to its weight.

An idealization of a vertically suspended spring is that it has no mass. If the ideal spring is suspended horizontally, we may assume that it slides on a frictionless horizontal surface. Essentially, Hooke's law is a first-order linear approximation to Hookean materials that are not stretched beyond its elastic limit. On the contrary, rubber is a non-Hookean material because its elasticity is stress dependent and sensitive to temperature. In Volume II of Feynman Lectures, he briefly clarifies that: “the general theory of elasticity, the atomic machinery that determine the elastic properties, and finally the limitations of elastic laws when the forces become so great that plastic flow and fracture occur (Feynman et al., 1964, section 38-1 Hooke’s law).”

Questions for discussion:

1. How would you explain that molecular forces are attractive?

2. How would you explain that molecular forces are repulsive?

3. How would you explain that molecular forces are proportional to the displacement?

The moral of the lesson: the molecular force at large distances is attractive and varies inversely as the seventh power of the distance, F = -k/r⁷, but it is repulsive at short distances; it is proportional to the displacement near the equilibrium position.

References:

1. Feynman, R. P. (1939). Forces in molecules. Physical Review, 56(4), 340-343.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley. 

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1966). The Feynman lectures on physics Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.

5. Huang, K. (1987). Statistical Mechanics (Second Edition). New York: John Wiley & Sons.

Section 12–2 Friction

(Drag force, F ≈ cv² / Static friction, F = μN / Sliding friction, F = μN)

In this section, Feynman discusses drag forces, static friction, and sliding friction (or kinetic friction).

1. Drag force, F ≈ cv²:

“…it is a remarkable fact that the drag force on an airplane is approximately a constant times the square of the velocity, or F ≈ cv² (Feynman et al., 1963, section 12–2 Friction).”

The drag force on an airplane is approximately a constant times the square of the velocity, F ≈ cv². According to Feynman, if the velocity of an airplane is extremely low, then the law of drag force needs to be modified because the drag force is directly proportional to the velocity (or approximately linear dependence). He explains that the drag force on an airplane flying through the air is not a simple law as it involves air molecules rushing over the wings, the swirling in the back, the changes going on around the fuselage, and many complications. Interestingly, it is possible for a drag force (or wave drag) to be proportional to the cube of the velocity, F ≈ kv³. Wave drag may occur as a result of pressure differences around a moving object in between two fluids such as air and water (Blazevich, 2017).

Feynman says that the drag force expressed in terms of F = cv² begin to fail at higher speeds not simply due to slight changes in the coefficient. He elaborates that the force on a wing depends upon the other wing and the rest of the plane. As an alternative, the drag force can be represented by the equation F = ½CrAv² where C is the drag coefficient, r is the density of the fluid, and A is the cross-sectional area of the object facing the fluid. Similarly, Feynman uses the formula C_D = F/[½rv²dl] for the drag force on a circular cylinder (Feynman et al, 1964, section 41–4). To be more precise, the drag force depends on the fluid’s viscosity and compressibility, the shape of the body, and the body’s inclination to the flow. The drag force equation can be derived to within a multiplicative constant by using dimensional analysis.

2. Static Friction, F = μN:

“… the frictional force is proportional to this normal force, and has a more or less constant coefficient; that is, F = μN, where μ is called the coefficient of friction (Feynman et al., 1963, section 12–2 Friction).”

Feynman distinguishes two kinds of drag forces that are due to fast movement in the air and slow movement in honey. He states that there is another kind of friction (dry friction or sliding friction) which occurs when one solid body slides on another. One may expect Feynman to explain that static friction is not really static because there are micro-displacements of atoms. More important, the formula F = μ_sN is approximately correct for static friction and it can be demonstrated by a simple experiment: we place a block of weight W on an inclined plane and measure the angle θ which the block begins to slide. This occurs when the component of the weight parallel to the plane (Wsin θ) is equal to the maximum static frictional force μ_sN (or μWcos θ). By equating the two expressions, we can deduce the static coefficient of friction, μ_s = tan θ.

The empirical law F = μ_sN has its limitations, such that it does not always work. Firstly, we should notice that when the inclined plane is tilted at the correct angle θ, the block does not slide steadily but in a halting fashion. The variations of coefficient μ_s are caused by different degrees of smoothness or hardness of the material, and possibly dirt, impurities, or oxides. Historically, there are two laws of static friction: (1) Static frictional force (μ_sN) is proportional to the normal reaction, N (Amontons’ first law). (2) Static frictional force is independent of the apparent area of contact, A (Amontons’ second law). However, the frictional force can be represented by the equation F = μN + kA, where kA is dependent on the area of contact between two surfaces.

3. Sliding friction, F = μN:

“…the friction to be overcome to get something started (static friction) exceeds the force required to keep it sliding (sliding friction), but with dry metals, it is very hard to show any difference (Feynman et al., 1963, section 12–2 Friction).”

Coulomb’s law of friction can be described as follows: the kinetic frictional force is independent of sliding speed. Feynman elaborates that the frictional force is not well understood and it is difficult to perform accurate experiments in friction. The apparent decreases of the kinetic frictional force at high speeds are often due to vibrations. Currently, physicists may explain that the kinetic frictional force decreases for materials such as steel, copper, and lead, but increases for the polymer Teflon (Besson et al., 2007). In general, frictional forces are influenced by many factors such as surface cleanliness, surface roughness, contact temperature, relative humidity, lubricant properties and presence of loose particles (Blau, 2008).

Many simply believe that the static friction exceeds the kinetic friction, but it is hard to tell any difference for dry metals. Feynman explains that the opinion may arise from experiences where some oils or lubricants are present, without using the term wet friction. If we try to have pure copper by cleaning and take every conceivable precaution, it is still difficult to determine the coefficient μ. It is even possible that two pieces of copper stick together if we tilt the apparatus to a vertical position. One reason for this behavior is that when the atoms in contact are of the same kind, there is no way for the atoms to “know” that they are in different pieces of copper. Physics teachers may conclude the section using the words of American Society of Mechanics Handbook (1992): “[u]niversal agreement as to what truly causes friction does not exist and much still remains to be done before a complete picture can emerge (p. 27).”

Questions for discussion:

1. Could a drag force be directly proportional to the velocity of an object as well as the square of the velocity of the same object simultaneously?

2. What are the idealizations, approximations, and limitations in conceptualizing static friction?

3. What are the idealizations, approximations, and limitations in conceptualizing kinetic friction?

The moral of the lesson: frictional forces (approximately μN) are due to molecular forces that cannot be satisfactorily explained by classical physics; we need quantum mechanics to understand them fully.

References:

1. American Society of Mechanics (1992). Friction, Lubrication, and Wear Technology. ASM Handbook, Vol. 18. Ohio: ASM International.

2. Besson, U., Borghi, L., De Ambrosis, A., & Mascheretti, P. (2007). How to teach friction: Experiments and models. American Journal of Physics, 75(12), 1106-1113.

3. Blau, P. J. (2008). Friction science and technology: from concepts to applications (2nd ed.). Boca Raton: CRC press.

4. Blazevich, A. J. (2017). Sports Biomechanics: The Basics: Optimising Human Performance. London: A & C Black.

5. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Section 12–1 What is a force?

(Circular definitions / Imprecise definitions / Idealizations and approximations)

In this section, Feynman discusses circular definitions of force, imprecise definitions of force, as well as idealizations and approximations involved in defining force.

1. Circular definitions:

“Now such things certainly cannot be the content of physics, because they are definitions going in a circle (Feynman et al., 1963, section 12–1 What is a force?).”

It is inappropriate to define constant momentum in terms of “the sum of the external forces is zero” and then define no net external forces in terms of “constant momentum.” Similarly, a definition of constant velocity as a result of “no force” and a definition of force based on “changing velocity” can be described as circular definitions. It is useless to have two definitions that contain two statements in which the premise is equivalent to the conclusion. We can find circular definitions in elementary textbooks and students would have difficulty learning science concepts (Arons, 1990). In some college textbooks, the concept of mass is defined as the resistance of a body to acceleration using the equation m = F/a, whereas force is the ‘cause’ of acceleration based on the same equation F = ma (Wong, Chu, Yap, 2014).

We can find circular definitions in the dictionary because words are always defined by some other words, which are eventually defined by some of the initially defined words within the same dictionary. For example, Feynman found that Webster defines time as a period, and period as time, which is not useful in explaining the concept of time (Feynman et al., 1963, section 5.1). The problem of circular definitions in the dictionary cannot be completely eliminated. Although dictionaries have shortcomings of circularity, this does not necessarily mean that they are definitely useless. In a sense, the problem of circularity in defining physical concepts is not simply a fallacy but a challenge, which cannot be easily resolved.

2. Imprecision definitions:

“If you insist upon a precise definition of force, you will never get it! (Feynman et al., 1963, section 12–1 What is a force?).”

The word force has some independent properties (e.g., magnitude, direction, and point of application), in addition to the law F = ma; but the specific independent properties of force were not completely described by Newton such that F = ma is an incomplete law. Feynman explains that one of the most important characteristics of force is that it has a material origin. More important, we do not have a precise definition of force because Newton’s Second Law of dynamics is not exact and it involves idealizations and approximations. Interestingly, Feynman clarifies that the concept of force is different from “gorce” that is related to the rate of change of position. In physics education research, the idea of gorce is reported as an alternative conception when students believe that “motion implies a force.”

Feynman is imprecise when he uses the word definition which may mean verbal definition, mathematical definition, or operational definition. In a lecture delivered at The University of Washington at Seattle, he mentions that “I think that extreme precision of definition is often not worthwhile, and sometimes it is not possible—in fact mostly it is not possible… (Feynman, 1998, p. 20). Essentially, he disagrees with philosophers who argue that words must be defined extremely precisely. Physics teachers may elaborate that the definition of force is not precise because of its rich culture. One may use Wilczek’s (2004) words: “physicists and mathematicians including notably Jean d’Alembert (constraint and contact forces), Charles Coulomb (friction), and Leonhard Euler (rigid, elastic, and fluid bodies) made fundamental contributions to what we now comprehend in the culture of force (pp. 11-12).”

3. Approximations and idealizations:

“…every object is a mixture of a lot of things, so we can deal with it only as a series of approximations and idealizations (Feynman et al., 1963, section 12–1 What is a force?).”

Newton’s second law of dynamics is formulated through a series of idealizations and approximations. Firstly, physicists may idealize a chair to be a definite thing in an ideal fashion or assume it to be a point object. Secondly, from an experimental perspective, we idealize the universe using Euclidean geometry and conduct land surveying by assuming a flat space-time instead of curved space-time. Next, Feynman clarifies that the mass of a chair can only be defined approximately because it is difficult to determine exactly which atoms belong to the chair, which atoms are air, which atoms are dirt, or which atoms are paints that belong to the chair—every object is a mixture of a lot of things. On the other hand, the forces on a single object also involve approximations of some kind in the real world.

The concepts of idealization and approximation may overlap in understanding various physics problems. In general, idealization involves constructing models or definitions that are relatively simple as compared to the real world, whereas approximation may refer to mathematical methods (e.g., numerical analysis) that are needed to solve physics problems. Importantly, in section 2.1, Feynman imagines the physical world is like a chess game being played by the gods and suggests three ways in understanding physical laws: idealization (or simplification), approximation (or imprecision) and exception (or violation). According to Feynman, an exception of the law F = ma is that the mass of an object is not constant if it is moving at high speeds. He also adds that F = ma is not really a definition because it is not always exactly true.

Questions for discussion:

1. How would you explain that there are problems of circularity in defining the concept of force?

2. What does Feynman mean when he says that we can never have a precise definition of force?

3. How do physicists conceptualize Newton’s second law of dynamics by using idealizations and approximations?

The moral of the lesson: we should be cognizant of problems of defining force that are related to circularity and imprecisions as well as the need of idealizations and approximations in conceptualizing Newton’s second law of dynamics.

References:

1. Arons, A. B. (1990). A Guide to Introductory Physics Teaching. New York: Wiley.

2. Feynman, R. P. (1998). The meaning of it all: Thoughts of a citizen scientist. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Wilczek, F. (2004). Whence the force of F = ma? I: culture shock. Physics Today, 57(10), 11-12.

5. Wong, C. L., Chu, H. E., & Yap, K. C. (2014). Developing a framework for analyzing definitions: a study of the Feynman lectures. International Journal of Science Education, 36(15), 2481-2513.

Section 11–7 Scalar product of vectors

(Algebraic definition / Geometric definition / Applications of scalar product)

In this section, Feynman discusses an algebraic definition of scalar product, a geometric definition of scalar product, and applications of scalar product.

1. Algebraic definition:

“… we now define the following quantity for any two vectors a and b: a.b = a_xb_x + a_yb_y + a_zb_z (Feynman et al., 1963, section 11–7 Scalar product of vectors).”

Feynman explains that the length of a step in space would be the same in any coordinate system. That is, the distance of a particular step r can be represented by r = √(x²+y²+z²) in a coordinate system and by r′ = √(x′²+y′²+z′²) in another coordinate system such that the distance r = |r| = |r′|. Thus, we can provide an algebraic definition of scalar product (or dot product) for any two vectors a and b: a.b = a_xb_x + a_yb_y + a_zb_z. An important property of scalar product is that the real number produced remains the same in any three mutually perpendicular axes. On the contrary, Feynman changes the order of presentation by first providing a geometric definition of scalar product followed by an algebraic definition in his Tips on Physics (Feynman et al., 2006).

To show that the scalar product a_xb_x+a_yb_y+a_zb_z is invariant in any coordinate systems, one may cite an important fact in which a.a, b.b, and c.c, are also invariant. Feynman suggests a method by expanding the equation: (a_x + b_x)² + (a_y + b_y)² + (a_z + b_z)² = (a_x′ + b_x′)² + (a_y′ + b_y′)² +(a_z′ + b_z′)². As a result, there will be sums of squares of the components of a and b as well as cross product terms such as a_xb_x and a_yb_y. However, it is relatively easy to prove that the scalar product is invariant in all coordinate systems by showing that a_xb_x + a_yb_y + a_zb_z = ½(|a + b|² – |a|² – |b|²). As another alternative, one may prefer to prove |b – a|² = |a|² + |b|² – 2a.b by using the cosine law of triangle (or cosine rule). In a sense, the scalar product of two vectors is tantamount to the cosine law.

2. Geometric definition:

“…geometrical way to calculate a.b, without having to calculate the components of a and b: a.b is the product of the length of a and the length of b times the cosine of the angle between them (Feynman et al., 1963, section 11–7 Scalar product of vectors).”

The scalar product of two vectors, a.b, can be geometrically defined as a product of the length of a and the length of b times the cosine of the angle between them. Feynman elaborates that we can choose a special coordinate system in which the x-axis lies along a such that the only component of a is a_x, which is the whole length of a. Thus, the equation a.b = a_xb_x+a_yb_y+a_zb_z is shortened to a.b = a_xb_x and this is simply the length of a times the component of b in the direction of a, that is, a(bcos θ). One may add that the scalar product of two vectors is closely related to the cosine law of triangle. Mathematically, the cosines of the angles (α, β, γ) between a vector and the three coordinate axes can be expressed as cos² α + cos² β + cos² γ = 1.

Succinctly, Feynman explains that if a.b = abcos θ is true in one coordinate system, it is true in all because a.b is independent of the coordinate system. Physics teachers should realize that the scalar product of two vectors is now expressed in polar coordinates instead of Cartesian coordinates. When the scalar product is expressed in polar coordinates, it is equal to the product of their lengths, multiplied by the cosine of the angle between them. One may also prefer Feynman’s additional explanation in his Tips on Physics: “[i]t is evident that since |A| cos θ is the projection of A onto B, A.B is equal to the projection of A onto B times the magnitude of B. Similarly, since |B| cos θ is the projection of B onto A, A.B also equals the projection of B onto A times the magnitude of A (Feynman et al., 2006, p. 28).”

3. Applications of scalar product:

“We have not yet defined work, but it is equivalent to the energy change, the weights lifted, when a force F acts through a distance s: Work=F.s (Feynman et al., 1963, section 11–7 Scalar product of vectors).”

Curiously, Feynman asks the question “[w]hat good is the dot product?” without first explaining the meaning of dot product. He admits that he has not defined work and states that it is equivalent to the energy change, that is, when a force F acts through a displacement s: Work = F.s. Importantly, the work done by a force is defined as the product of the force and the parallel distance over which it acts. In short, the term work is misleading to students and it may be rephrased as “force-displacement product” or “dot product of force and parallel distance”. This is in contrast to the term torque that is a “cross product of force and perpendicular distance. Note that a torque is dependent on the magnitude of the force applied and its perpendicular distance from the axis of rotation.

Feynman ends the chapter by discussing unit vector whose dot product with itself is equal to unity. The dot products of unit vectors are summarized as i.i = 1, i.j = 0, j.j = 1, i.k = 0, j.k = 0, and k.k = 1. By applying definitions of unit vectors, we can represent physical quantities such as forces by writing their components in the form of a = a_xi + a_yj + a_zk. Historically, the concepts of scalar product and vector product were developed by Hamilton and Grassmann independently in the 1840’s. The word vector was first coined by Hamilton and it can be found in Hamilton’s (1853) works such as Lectures on Quaternions. The usual form of a quaternion is x = x₀ + x₁i + x₂j + x₃k, however, i.i = -1, j.j = -1, k.k = -1, i.j = 0, j.k = 0, and k.i = 0.

Questions for discussion:

1. How would you define a scalar product of two vectors algebraically?

2. How would you explain a scalar product of two vectors geometrically?

3. Why are scalar products of two vectors applicable in physics?

The moral of the lesson: a scalar product of two products can be defined algebraically and geometrically, and it can be applied in physics concepts such as work.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.