Section 28–2 Radiation

(Unit vector / Inverse distance dependence / x-component of acceleration)

 

The three interesting concepts discussed in this section are the unit vector in Feynman’s formula for the electric field, its property of inversely proportional to distance, and the x-component of acceleration in the simplified formula.

 

1. Unit vector:

“What this term says is: look at the charge and note the direction of the unit vector (we can project the end of it onto the surface of a unit sphere) (Feynman et al., 1963, section 28–2 Radiation).”

 

Feynman defines the unit vector e_r′ as pointed toward the apparent position of the charge. In addition, the end of e_r′ may move like a curve so that its acceleration has two components. In spherical coordinates, a vector r can be represented as follows: r = r × rˆ(q, f) in which r is the magnitude of the vector r, rˆ is the unit vector, q is the polar angle, and f is the azimuthal angle. Specifically, the vector r may be described as the line of sight vector if it is from the perspective of an observer. Perhaps it is confusing to describe the transverse component as going up and down because it is a sidewise component that could be going east-west, north-south, or the resultant of these two directions.

 

Feynman claims that it is easy to see the wigglings of e_r′ of a given source would vary inversely as the distance when it is moving farther away. Although Feynman considers the unit vector to be easy, but the rules of unit vector can be complicated. For example, it is not simple for some students to deduce drˆ/dt = (drˆ/dq)(dq/dt) + (drˆ/df)(df/dt) = (icos q cos f + jcos q sin f − ksin q)(dq/dt) + (−i sin q sin f + j sin q cos f)(df/dt) = qˆ(dq/dt) + fˆ(sin q)(df/dt). Furthermore, a = (d²r/dt² – r[dq/dt]² – r[df/dt]²sin² q)rˆ + (r[d²q/dt²] + 2[dr/dt][dq/dt] – r[df/dt]²sin q cos q)qˆ + (r[d²f/dt²] sin q + 2[dr/dt][df/dt]sin q + 2r[dq/dt][df/dt]cos q)fˆ. Worst still, this unit vector is dependent on the retarded time r′/c and retarded distance r′. It is elaborated in Chapter 34 of Volume I of The Feynman Lectures.

 

2. Inverse distance dependence:

“To discuss these phenomena, we must select from Eq. (28.3) only that piece which varies inversely as the distance and not as the square of the distance (Feynman et al., 1963, section 28–2 Radiation).”

 

Feynman mentions that the electric field at a point is inversely proportional to the distance from the charge, but the expression of the electric field is given as a law and it will be learnt in detail next year. In vol II, he says: “[i]t turns out that we won’t quite make it—that the mathematical details get too complicated for us to carry through in all their gory details (Feynman et al., 1964, chapter 21).” He simplifies the situation by suggesting some charges are moving only a small distance at a slow rate. This tiny motion is effectively at a constant distance implies that the unit vector can be represented using Cartesian coordinates, rˆ = (x/r, y/r, z/r), in which r = Ö(x² + y² + z²). In essence, the charged object must be moving at a slow speed radially such that r can be considered as a constant (thus d²rˆ/dt² = d²(x/r)/dt² = a_x/r).

 

In vol II, Feynman uses a “bullet” analogy to explain how an electric field of a point charge is inversely proportional to the square of the distance. In his words, “[i]f the gun is enclosed in a surface, whatever size and shape it is, the number of bullets passing through is the same—it is given by the rate at which bullets are generated at the gun… (Feynman et al., 1964, section 4-5).” Similarly, Zangwill (2012) explains the inverse distance dependence using an expanding annular ring. In this case, the total electric flux captured by the ring is E_a2πRdR and the constant value of the total electric flux leads to E_a(R) µ 1/R. One should be cognizant that many forces in physics such as the nuclear force and molecular force do not obey the inverse square law.

 

3. x-component of acceleration:

“…r is practically constant, the x-component of d²e_r′/dt² is simply the acceleration of x itself at an earlier time divided by r, and so finally we get the law we want, which is E_x(t) = (−q4πϵ₀c²/r)a_x(t−r/c). (Feynman et al., 1963, section 28–2 Radiation).”

 

According to Feynman, if a charged object is moving in a very small motion and it is laterally displaced by the distance x, then the unit vector e_r′ is displaced by x/r. The x-component of d²e_r′/dt² is simply the acceleration of x at an earlier time divided by r provided r is constant. This also implies the acceleration of r(t) or possibly z(t) is zero. On the other hand, one may consider the distance x(t) to be close to zero because the charged object is moving at a relatively slow speed and it is far away from the observer. In other words, the distance x that is laterally displaced, is possibly very short in comparison to r such that we can use the small-angle approximation formula q @ sin q = x/r.

 

Feynman elaborates that Eq. (28.5) is the complete and correct formula for radiation, and even relativity effects are all contained in this formula. This statement is not completely correct because Feynman’s formula for the electric field is a special case of Jefimenko’s equations. One may elaborate that Lorentz’s factor and the constant speed of light in all inertial frames are hidden in Maxwell’s equations. In an article titled Why is Maxwell’s Theory so hard to understand, Dyson (1999) writes, “[w]e may hope that a deep understanding of Maxwell’s theory will result in dispersal of the fog of misunderstanding that still surrounds the interpretation of quantum mechanics.” In short, Maxwell’s field theory does not only contain relativity effects, but the prototype of quantum field theory.

 

Review Questions:

1. Would you consider the unit vector to be simple and describe its transverse component as only going up and down?

2. How would you explain the electric field is inversely proportional to the distance?

3. Is Eq. (28.5) the complete and correct formula for radiation and only relativity effects are all contained in this formula?

 

The moral of the lesson: if a charge is moving at a slow speed and it is laterally displaced by the distance x(t), then the angle that the unit vector e_r′ is displaced is x/r, and since r is practically constant, we have E_x(t)= (−q4πϵ₀c²/r)a_x(t−r/c).

 

References:

1. Dyson, F. (1999). Why is Maxwell’s theory so hard to understand? In James Clerk Maxwell Commemorative Booklet, Fourth International Congress Industrial and Applied Mathematics, Edinburgh, Scotland.

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. Zangwill, A. (2012). Modern Electrodynamics. New York: Cambridge University Press.

Section 28–1 Electromagnetism

 (Retarded Coulomb field / Correction term / Self-force)

 

The three interesting concepts discussed in this section are retarded coulomb field, its correction term, and self-force.

 

1. Retarded Coulomb field:

“Take the first term, E = −qe_r′/4πϵ₀r′². That, of course, is Coulomb’s law, which we already know: q is the charge that is producing the field (Feynman et al., 1963, section 28–1 Electromagnetism).”

 

Feynman says that the first term is Coulomb’s law in which q is the charge that is producing the field. In a sense, this is imprecise because the first term could be known as Retarded Coulomb field because it includes the retarded time, r′/c, or the time it takes for the influence to move from the charge to the point P at speed c. Note that Feynman uses the phrase delayed Coulomb field subsequently, but the notation of the term −qe_r′/4πϵ₀r′² should be improved to denote the delay effect. Some may prefer Jackson’s notation that expresses this term as E = (q/4pe₀)(Ȓ/R²)_ret. The idea of time delay may be illustrated by Feynman’s explanation of electric field using the analogy of two floating corks in water (Chapter 2) where water waves are analogous to electromagnetic waves.

Feynman explains that Coulomb’s law is wrong because influences cannot travel faster than the speed of light. Alternatively, one may clarify that Coulomb’s law is an idealization because we need to assume the “charges” are point objects and they are stationary instead of moving continuously. Furthermore, this law is not applicable for very short distances (less than 10⁻¹⁶ m) and we are unable to experimentally prove that it holds for very large (astronomical) distances. To give a better idea of field, Wilczek (2015) writes: “…in applications to fundamental physics, where quantum fluctuations are important, it becomes problematic because both forces and positions fluctuate. It can be salvaged as an approximate notion by doing some averaging over time and space (p. 355).”

2. Correction term:

“It suggests that we should calculate the delayed Coulomb field and add a correction to it, which is its rate of change times the time delay that we use. (Feynman et al., 1963, section 28–1 Electromagnetism).”

 

Feynman explains that the second term is as though nature was trying to allow for the fact that the effect is retarded. In a sense, this explanation is potentially misleading because the three terms of equation (28.3) have the same retarded effect. Many may prefer Jackson’s (1999) use of subscript “ret” in Feynman’s expression for the electric field is “E = (q/4pe₀) {[Ȓ/kR²]_ret + (¶/c¶t)[Ȓ/kR]_ret -(¶/c²¶t)[v/kR]_ret} (p. 284).” More important, Feynman adds that it is easy to show the first two terms vary inversely as the square of the distance in the next section. That is, we can use chain rule dy/dt = (dy/dx)(dx/dt) and in this case, d(1/R)/dt = d(1/R)/dR × dR/dt. It is simple to realize that d(1/R)/dR varies inversely as the square of the distance R.

In volume II, footnote 1 of chapter 21 states: “1. The formula was first published by Oliver Heaviside in 1902. It was independently discovered by R. P. Feynman, in about 1950, and given in some lectures as a good way of thinking about synchrotron radiation.” This footnote on Feynman’s expression for the electric field or equation (28.1) is incorrect because Heaviside’s formula is an expression for the magnetic field. However, Feynman’s expression for the electric field is equivalent to Heaviside’s expression for the magnetic field (Jackson’s 1999). Furthermore, footnote 2 of chapter 21 suggests readers not to derive Feynman’s expression for the electric field. On the contrary, this is a problem for students in Jackson’s (1999) Classical Electrodynamics by using Jefimenko’s equations. (Feynman’s expression for the electric field is a special case of Jefimenko’s equations.)

 

3. Self-force:

“…we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero (Feynman et al., 1963, section 28–1 Electromagnetism).”

 

According to Feynman, when we try to calculate the field from all the charges including the charge, we get into trouble, e.g., dividing something by the distance, which is zero. In other words, we have difficulties in calculating the self-force of a charge that seems to be infinity. Feynman adds that the problem of handling the electric field which is generated by the same charge was not yet solved. In volume II, Feynman elaborates that “…the infinity arises because of the force of one part of the electron on another—because we have allowed what is perhaps a silly thing, the possibility of the ‘point’ electron acting on itself (Feynman et al., 1964, section 28–4 The force of an electron on itself).” Recently, Gralla, Harte, and Wald (2009) determine the self-force by assuming the charge (q) and total mass (m) approach zero, and q/m reaches a well-defined limit.

 

Historically, Wheeler and Feynman propose that point charges interact only with other charges, but the interaction is half through the advanced waves and half through the retarded waves. In his Nobel lecture, Feynman (1965) says that “the idea seemed so obvious to me and so elegant that I fell deeply in love with it. And, like falling in love with a woman, it is only possible if you do not know much about her, so you cannot see her faults…” Interestingly, when Feynman was a graduate student in Princeton, he shared this idea in a technical talk and Einstein politely said: “I find only that it would be very difficult to make a corresponding theory for gravitational interaction … Since we have at this time not a great deal of experimental evidence, I am not absolutely sure of the correct gravitational theory (Feynman, 1985, p. 80).”

 

Review Questions:

1. Does the first term of the Heaviside-Feynman expression of electric field refers to Coulomb’s law or retarded Coulomb field?

2. How would you explain the correction term for the retarded Coulomb field (just like Feynman)?

3. Is the problem of self-force still not completely solved today?

 

The moral of the lesson: Coulomb’s law is an idealization not only because we need to assume the “charges” are point objects and they are stationary, but this law has not included the time-delay effect of influences.

 

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. Gralla, S. E., Harte, A. I., & Wald, R. M. (2009). Rigorous derivation of electromagnetic self-force. Physical Review D, 80(2), 024031.

3. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press. 

Section 27–7 Resolving power

(Optical resolution / Rayleigh’s criterion / Limitations of geometrical optics)

 

The three interesting concepts discussed in this section are optical resolution, Rayleigh’s criterion, and the limitations of geometrical optics.

 

1. Optical resolution:

“The general rule for the resolution of any optical instrument is this: two different point sources can be resolved only if one source is focused at such a point that the times for the maximal rays… (Feynman et al., 1963, section 27–7 Resolving power).”

 

Feynman explains resolving power by providing the general rule for optical resolution that is related to resolving two different point sources such as looking at a bacterium. However, there could be more discussions on the definition of resolving power and the resolution of an optical instrument. Firstly, the resolving power of a microscope or a telescope is its ability to separate the images of two objects and it can be expressed in terms of angular resolution as q = 1.22 l/D in which D is the diameter of the aperture. Next, the resolving power of a spectroscope or diffraction grating is its ability to separate the wavelengths and it can be expressed as l/Dl. In general, the better the resolving power (smaller resolving power) implies the better the optical resolution (or the smaller size the instrument can resolve).

 

Feynman was aware of the resolution or resolving power of electron microscopes. In his lecture titled There’s plenty of room at the bottom, Feynman (1959) says: “I would like to try and impress upon you while I am talking about all of these things on a small scale, the importance of improving the electron microscope by a hundred times. It is not impossible; it is not against the laws of diffraction of the electron (p. 124).” Interestingly, Feynman poses the challenge of a more powerful electron microscope: “there are theorems which prove that it is impossible, with axially symmetrical stationary field lenses, to produce an f-value any bigger than so and so; and therefore the resolving power at the present time is at its theoretical maximum. But in every theorem there are assumptions. Why must the field be symmetrical? (p. 126).”

2. Rayleigh criterion:

“A corresponding formula exists for telescopes, which tells us the smallest difference in angle between two stars that can just be distinguished (Feynman et al., 1963, section 27–7 Resolving power).”

 

According to Feynman, a corresponding formula exists for telescopes, which tells us the smallest difference in angle between two stars that can just be distinguished. In section 30-4, Feynman states the resolving power of a telescope as θ = 1.22λ/L, where L is the diameter of the telescope. Essentially, he considers Rayleigh criterion to be the limit of resolving power whereby two point-sources are just resolved when the central maximum of one image coincides with the first minimum of the other. Currently, Rayleigh’s criterion is no longer considered to set the limit of resolving power. For example, Born and Wolf (1980) write: “[w]ith other methods of detection (e.g. photometric) the presence of two objects of much smaller angular separation than indicated by Rayleigh’s criterion may often be revealed (p. 418).”

 

Feynman mentions that if the distance of separation of the two points is D and if the opening angle of the lens is θ, then the inequality t₂ − t₁ > 1/ν is exactly equivalent to D > λ/nsin θ and suggests the best resolution is approximately the wavelength of light. It seems that Feynman would describe the criterion to be a rough idea. (In Chapter 30, a footnote is stated: This is because Rayleigh’s criterion is a rough idea in the first place.) However, it is not obvious nor practical to operationalize the limit of resolving power (t₂ − t₁ > 1/ν) as stated by Feynman. For instance, it is more practical to resolve binary stars using Dawes’ limit that depends on the difference in brightness between the binary star components and the observer’s visual acuity instead of simply the optical resolving power of the telescope.

 

3. Limitations of geometrical optics:

“…we still could not see two points that are too close together because of the limitations of geometrical optics, because of the fact that least time is not precise (Feynman et al., 1963, section 27–7 Resolving power).”

 

Feynman explains that we cannot keep on magnifying the image because of the limitations of a microscope. He adds that this is due to the limitations of geometrical optics because of the fact that least time is not precise. However, many may expect Feynman to explain the limitations of geometrical optics that are related to the diffraction and interference of light waves. That is, wave properties of light cause difficulties to see two objects or light sources that are very close together. It implies that even we can compensate for aberrations, we should not expect to achieve perfectly sharp images because of the diffraction limit.

 

Feynman seems pessimistic to suggest that if “the difference in time is less than about the period that corresponds to one oscillation of the light, then there is no use improving it any further” (the end of the previous section). Currently, there are many ways to achieve a better resolution that is not limited by the diffraction effects (Tsang, Nair, & Lu, 2016). Physicists can use quantum optics, quantum metrology, and statistical analysis to provide a better estimate of the separation of two light sources. Historically, Rayleigh’s criterion was not rigorously proved and it was based on Huygen’s wave theory. In Sparrow’s (1916) words, “[a]s originally proposed, the Rayleigh criterion was not intended as a measure of the actual limit of resolution, but rather as an index of the relative merit of different instruments (p. 76).”

 

Review Questions:

1. Is the best resolution approximately the wavelength of light or the size of a molecule (See Hell’s Nobel lecture)?

2. Would you consider Rayleigh’s criterion of resolution to be a rough idea?

3. What are the limitations of geometrical optics?

 

The moral of the lesson: we may not be able to resolve two light sources that are close together because of the diffraction and interference of light waves.

 

References:

1. Born, M. & Wolf, E. (1980). Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (6th ed.). Oxford: Pergamon.

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. Hell, S. W. (2015). Nanoscopy with focused light (Nobel Lecture). Angewandte Chemie International Edition, 54(28), 8054-8066.

4. Sparrow, C. M. (1916). On spectroscopic resolving power. The Astrophysical Journal, 44, 76-86.

5. Tsang, M., Nair, R., & Lu, X. M. (2016). Quantum theory of superresolution for two incoherent optical point sources. Physical Review X, 6(3), 031033.

Section 27–6 Aberrations

 (Spherical aberration / Chromatic aberration / Comatic aberration)

 

In this section, Feynman discusses spherical aberration, chromatic aberration, and comatic aberration (or coma).

 

1. Spherical aberration:

“This effect is called spherical aberration, because it is a property of the spherical surfaces we use in place of the right shape (Feynman et al., 1963, section 27–6 Aberrations).”

 

According to Feynman, a real lens having a finite size will exhibit aberrations, for example, spherical aberration is a smear in an image. The word spherical is used because spherical aberration is related to the spherical surfaces of the lenses that enlarge an image imperfectly. Simply phrased, an aberration is an image error of an optical system that may manifest as an unclear or distorted image. Spherical aberration is classified as a type of monochromatic (or quasi-monochromatic) aberrations, however, other types of monochromatic aberrations include coma, astigmatism, field curvature, and distortion. More important, one should emphasize that spherical aberration is observable even for objects that are located on the optical axis (or principal axis).

 

Feynman elaborates that the spherical aberration could be remedied by re-forming the shape of the lens surface or using several lenses arranged so that the aberrations of the individual lenses tend to cancel each other. One may include more methods to resolve this aberration such as using an aperture stop or computerized lens design. In short, the aperture stop can affect the amount of light closer to the optical axis that passes through the lens. Perhaps Feynman should mention that computerized lens design is a useful tool that was used by manufacturers worldwide in the early 1960s (when this lecture was delivered). Currently, there are more sophisticated computer programs that help to design and analyze more complicated optical systems (Hecht, 2002).

 

2. Chromatic aberration:

“So if we image a white spot, the image will have colors, because when we focus for the red, the blue is out of focus, or vice versa. This property is called chromatic aberration (Feynman et al., 1963, section 27–6 Aberrations).”

 

According to Feynman, another fault of the lens is its refractive index which is color-dependent, and thus, light of different colors travels at different speeds in a glass. That is, a white spot has chromatic aberration in the sense that its image has different colors. Chromatic aberration is also known as “color fringing” or “splitting of light” because the lens is unable to let colored light rays meet at the same point in the focal plane. Specifically, lens dispersion is observed as a result of a higher refractive index for light rays that have shorter wavelengths. In other words, the image appears blurred with colored edges because light of different colors reaches different points along the optical axis.

 

Instead of providing a specific solution to compensate for chromatic aberration, Feynman asks how careful do we have to be to eliminate aberrations. Then, he says that the theory of geometrical optics does not work here and the principle of least time is only an approximation. Perhaps Feynman should discuss how achromatic lenses can resolve the chromatic aberration from the perspective of Fermat’s principle of least time. For example, one may explain to what extent the light path of each ray has the same length by having a good design of lens surface with the appropriate refractive index. In essence, Fermat’s principle of least time is a first-order approximation in the sense that the optimum light path could have the longest-time or shortest-time provided all nearby paths take approximately the same time (δT = 0).

 

3. Comatic aberration: “If the object is off the axis, then the focus really isn’t perfect anymore, when it gets far enough off the axis (Feynman et al., 1963, section 27–6 Aberrations).”

 

Feynman says that the focus isn’t perfect anymore if the object is located off the optical axis. (Strictly speaking, the focus isn’t perfect even if the object is located on the optical axis depending on its size and colors.) He elaborates that the image will usually be quite crude, and there may be no place where it focuses well. One may clarify that this image error is also known as coma or comatic aberration. In short, coma is a monochromatic aberration that occurs for an object located a distance from the optical axis; the light rays reach points on the focal plane that are farther from the optical axis. A good example is the appearance of comet-shaped stars when they are located at an angle to the optical axis of a telescope.

 

Feynman mentions that the optical designer tries to remedy aberrations by using many lenses to compensate for each other’s errors. Then, he explains that if we have arranged the time difference for different light rays is less than about a period, there is no use going any further. One may add that comatic aberration can be compensated by using an aperture stop at the proper location. However, if the time difference for different light rays is less than about a period, then there would be interference between the light rays such that it is more difficult to improve the resolution of the image. This is a limitation of geometrical optics because of the diffraction and interference of light waves.

 

Review Questions:

1. How would you define the concept of spherical aberration?

2. How would you resolve the problems of chromatic aberration?

3. Do you agree with Feynman when he says that the focus isn’t perfect anymore if the object is located off the optical axis?

 

The moral of the lesson: aberrations (image errors) are due to the faults of lenses (spherical surface and refractive index) that cause light rays not to meet at the focal point and at the same time (Fermat’s principle of least time).

 

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. Hecht, E. (2002). Optics (4th ed.). San Francisco, Addison-Wesley.