Section 34–2 Finding the “apparent” motion

(Line-of-sight co-ordinate / Curtate cycloid / sharp cusp)

 

In this section, Feynman discusses the concept of line-of-sight co-ordinate, curtate cycloid, and sharp cusp that help to explain the emission of synchrotron radiation. This section could also be titled as “Finding the retarded lateral motion” instead of “Finding the apparent motion” because the lateral motion (or distance) matters more than the horizontal distance.

 

1. Line-of-sight coordinate:

“In this way we get a new motion, in which the line-of-sight coordinate is ct, as shown at the right… The point is that the horizontal (i.e., line-of-sight) distance now is no longer the old z, but is z+cτ, and therefore is ct (Feynman et al., 1963, p. 34–3).”

 

It may not be clear why Feynman defines horizontal (line-of-sight) distance in terms of ct = z+cτ. However, the main purpose is to deduce the retarded field of a relativistic electron using the second derivative of its retarded (lateral) position (x¢ or y¢). Pedagogically, it is useful to draw a graph of lateral distance with respect to time (or line-of-sight distance) to illustrate the variation of d²e_r′/dt². That is, we can use the large apparent transverse acceleration at a certain point to explain the synchrotron radiation. It may be worth mentioning that the retarded field does not depend on z (except the time delay) since it is practically constant. In essence, the rule is “we take the actual motion, translate it backwards at speed c, and that gives us a curve whose curvature measures the electric field (Feynman et al, 1963, section 34-5).”

 

We take the actual motion of the charge (shown at left) and imagine that as it is going around it is being swept away from the point P at the speed c (there are no contractions from relativity or anything like that; this is just a mathematical addition of the cτ) (Feynman et al., 1963, p. 34–2).

 

Note that the analysis is simplified such that it does not involve Lorentz contraction, but there is a geometrical effect that is similar to Doppler effect. During a circular motion, an electron is moving away from the observer over one half of a period, but it is moving towards the observer over the other half of the period. As a result, we expect a red shift or blue shift in the radiation depending on whether the electron is moving away or towards the observer. The most intense radiation occurs when many electrons are moving closely behind the photons that are emitted at the earlier (retarded) times. One may also idealize two photons emitted by an electron at two different positions and times to reach the observer at the same time (Kim, 1989).

 

“So the final answer is: in order to find the electric field for a moving charge, take the motion of the charge and translate it back at the speed c to “open it out”; then the curve, so drawn, is a curve of the x′ and y′ positions of the function of t (Feynman et al., 1963, p. 34–3).”

 

Feynman explains that there is an “open it out” effect in the curve of lateral distance with respect to time. However, the horizontal distance (= ct) is not the distance between the charge and observer because it is continuously increasing when an electron returns to its original position. Furthermore, there is a time squeezing effect (Kim, 1989) due to the electron “makes a U-turn to catch up” with the photons that were emitted at earlier times. In other words, the “time of photon emission” appears to be compressed when the electron is moving toward the observer, and it results as a sharp pulse of radiation. Perhaps it is good to include a diagram (see below) that shows the distortion of the curve of the x′ from a sinusoidal curve. Interestingly, Feynman suggests, “imagine that this whole ‘rigid’ curve moves forward at the speed c through the plane of sight.” 

Note: In section 34-3, Feynman uses the phrase piling up effect to describe Fig. 34-3. 

 

Source: Patterson, 2011

2. Curtate cycloid:

“Therefore, when we translate the motion back at the speed of light, that corresponds to having a wheel with a charge on it rolling backward (without slipping) at the speed c; thus we find a curve which is very close to a cycloid—it is called a hypocycloid (Feynman et al., 1963, p. 34–3).”

 

Feynman is correct in using the term cycloid, but the curve was revised to be called a curtate cycloid in the latter edition. In general, a cycloid is defined as a curve formed by a point on the circumference of a circle or wheel as it rolls along a straight line without slipping. To be specific, a curtate cycloid is the curve formed by a fixed point inside the rolling circle without slipping (See below). It is also known as contracted cycloid because the distance from the fixed point to the center of a rolling circle is shorter (corresponds to a lower speed) whereby the curve is formed by the point inside (instead of on) the circle. However, the x-axis of the graph could be ct instead of t, i.e., it is just a scaling of time-axis as a useful geometric representation for an electron moving at almost the speed of light.

Source: Weisstein, Eric W. "Curtate Cycloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurtateCycloid.html

A hypocycloid is a curve traced by a fixed point on a smaller circle that rolls within a larger circle instead of a straight line (See below). Although the term hypocycloid is incorrectly used in this chapter, we may still apply it in the study of electromagnetic waves. For example, it has been shown that the path traced by an electric field vector in space as an electromagnetic wave propagates can be described by a hypocycloid. This can be useful in the analysis and modeling of complex electromagnetic systems, such as antennas and waveguides (Singh, Prasad, & Ojha, 2003). In modern physics, the concept of hypocycloid may be applicable in superstring compactifications of Calabi-Yau threefolds (Bartolo & Cogolludo-Agustín, 2017).

Source: Weisstein, Eric W. "Hypocycloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hypocycloid.html

3. Sharp cusp:

“If the charge is going very nearly at the speed of light, the “cusps” are very sharp indeed; if it went at exactly the speed of light, they would be actual cusps, infinitely sharp. “Infinitely sharp” is interesting; it means that near a cusp the second derivative is enormous (Feynman et al., 1963, p. 34–3).”

 

The term cusp may be incorrectly used in the context of normal cycloid, curtate cycloid, or prolate cycloid (see below). Firstly, one may describe a curve of the x′ with respect to time as a normal cycloid if there is a cusp in the curve. However, the cusp of the normal cycloid of an electron is an idealized concept because it is impossible for the electron to move at the speed of light. Thus, some may consider the term curtate cycloid because the electron moves slower than the speed of light. Perhaps mathematical physicists prefer using higher curvature (instead of cusp) to explain synchrotron radiation because a curtate cycloid does not really have any cusp or vertical tangent.

Source: Weisstein, Eric W. "Prolate Cycloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProlateCycloid.html

Currently, undulators and wigglers are inserted into a synchrotron to control the beam and generate more radiation. In short, electrons are oscillating perpendicularly to their circular motion due to the alternating magnetic fields of the two inserted devices. Viewing the circular motion from the top, the electrons may oscillate in the east-west direction or north-south direction depending on their location. Thus, another option to model synchrotron radiation could be a prolate cycloid with retrograde loops because an electron reverses its horizontal and vertical direction in the real world. Using prolate cycloid, we may explain that more photons are squeezed into a shorter period of time such that the density of photons is being increased and it results in a very short and higher-intensity pulse of radiation.

Review Questions:

1. Do you agree with Feynman’s use of the term horizontal distance or line-of-sight co-ordinate? What is the physical meaning of horizontal distance?

2. Would you explain synchrotron radiation using the concept of general cycloid, curtate cycloid, or prolate cycloid?  

3. Does a curtate cycloid has sharp cusps?  

 

The moral of the lesson: By finding the lateral motion of an electron, it helps to explain how more photons are squeezed into a shorter period of time such that it results in a very short and higher-intensity pulse of radiation.

 

References:

1. Bartolo, E. A., & Cogolludo-Agustín, J. I. (2017). On the topology of hypocycloids. arXiv preprint arXiv:1703.08308.

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. Kim, K. J. (1989). Characteristics of Synchrotron Radiation, Physics of Particle Accelerators. American Institute of Physics (AIP), 184. 565-632.

4. Patterson, B. D. (2011). A simplified approach to synchrotron radiation. American Journal of Physics, 79(10), 1046-1052.

5. Singh, V., Prasad, B., & Ojha, S. P. (2003). Theoretically obtained dispersion characteristics of an annular waveguide with a guiding region cross section bounded by two hypocycloidal loops. Microwave and Optical Technology Letters, 37(2), 142-145.