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