Section 27–4 Magnification

 (Lateral magnification / Longitudinal magnification / Angular magnification)

 

In this section, we can find mathematical expressions that are related to lateral magnification, longitudinal magnification, and angular magnification.

 

1. Lateral magnification:

“Equation (27.15) is the famous lens formula; in it is everything we need to know about lenses: It tells us the magnification, y′/y, in terms of the distances and the focal lengths (Feynman et al., 1963, section 27–4 Magnification).”

 

Feynman defines the magnification of an optical system as y′/y that is in terms of an object height and image height as well as using xx′ = f² that is in terms of extrafocal distances (x and x′) and the focal length. However, the optical system is idealized to have only a focal length (instead of focal lengths: xx′ = ff′) and no aberration (see section 27.6). Feynman could have distinguished the concept of magnification using the terms lateral magnification and longitudinal magnification. Specifically, we can define lateral (or transverse) magnification as the ratio of the image height to the object height of the optical system. The lateral magnification would have a negative value if we consider the image is inverted with respect to the object.

 

In order to understand the formation and magnification of an image better, Feynman state two useful facts: 1. parallel rays from one side of the lens would proceed toward the focal point that is at a distance f from a lens. 2. any ray that passes the focal point on one side of a lens will emerge parallel to the optical axis on the other side. Importantly, there is one more useful fact: any ray that passes through the optical center of the lens will continue to move in a straight line. Furthermore, the first two conditions are applicable to a thick lens by including the concept of principal planes. This concept is also applicable to an optical system that has more than one lens (see section 27.5).

 

2. Longitudinal magnification:

“We leave it to the student to demonstrate that if we call s = x+f and s′ = x′+f, Eq. (27.12) is the same as Eq. (27.16) (Feynman et al., 1963, section 27–4 Magnification).”

 

Feynman mentions that y′/y = x′/f = f/x Eq. 27.15 is the famous lens formula and it has everything we need to know about lenses. Historically, the concept related to xx′ = f² appeared in Newton’s Opticks (Hecht, 2002) and thus, it is known as the Newtonian lens formula. Strictly speaking, the equation does not contain everything about lenses (e.g., aberration) and Feynman could have clarified how the lateral magnification of a lens is connected to the image distance s′ and object distance s (as well as x′ and x). Importantly, it is not trivial to explain that the extrafocal distances x and x′ are measured from the principal planes instead of the optical center. Furthermore, it is beneficial to explain that the longitudinal (or axial) magnification X is defined to be equal to x′/x (or X = Dx′/Dx = s′/s).

 

Feynman leaves it to his students to demonstrate that s = x+f and s′ = x′+f Eq. 27.12 are the same as xx′ = f² Eq. 27.16. However, it is worthwhile to let students demonstrate the concept of longitudinal magnification by differentiating the Newtonian lens formula. Firstly, we may express xx′ = f² as x′ = f²/x and differentiating x′ with respect to x, we have Dx′/Dx = f²(-x^-2) = -f²/x². Secondly, using xx′ = f² and y′/y = x′/f = f/x Eq. 27.15, we can get x′ = f(y′/y) and x = f/(y′/y). Thus, x′/x = f(y′/y) ¸ f/(y′/y) = (y′/y)² and it means that the longitudinal magnification X = x′/x is equal to the square of lateral magnification Y = y′/y. In a sense, the longitudinal magnification is about to what extent the image “becomes” larger because it appears farther.

 

3. Angular magnification:

“Now we may derive a lens formula. Using the similar triangles PVU and TXU, we find y′/f = y/x (Feynman et al., 1963, section 27–4 Magnification).”

 

To derive the Newtonian lens formula, Feynman uses the similar triangles PVU and TXU to get y′/f = y/x Eq. 27.13 and from similar triangles SWR and QXR to get y′/x′ = y/f Eq. 27.14. However, the angular magnification of a magnifying device is defined as M_a = q′/q or M_a = tan q′/tan q in which tan q = y/x (Note: y/x appears in Eq. 27.13). Simply phrased, angular magnification is the ratio of the angle subtended by an image (using a magnifying device) to the angle subtended by an object (naked eye). This is an important concept that is used to specify the magnification of telescopes and microscopes. (Feynman briefly mentions telescopes and microscopes in the next section without further elaboration.)

 

Some may understand the concept of magnification better by comparing lateral magnification, longitudinal magnification, and angular magnification. In short, we can define angular magnification, lateral magnification, longitudinal magnification as M_a = tan q′/tan q = (y′/x′)/(y/x), Y = y′/y, and X = x′/x respectively. The three magnifications can be related as M_a = (y′/x′)/(y/x) = (y′/y)/(x′/x) = Y/X. That is, we have Lateral magnification = Longitudinal magnification × Angular magnification (Y = X × M_a). On the other hand, it can be shown that the angular magnification of a lens is the reciprocal of its linear magnification (M_a = 1/Y). By assuming the object is small and it is placed on the optical axis, we have M_a = tan q′/tan q = (h/s′)/(h/s) = 1/(s′/s) that is valid for paraxial rays (Landsberg, 2000).

 

Review Questions:

1. How would you define the concept of lateral (or linear) magnification?

2. Does the famous lens formula contain everything we need to know about lenses, such as longitudinal magnification?

3. How would you explain the concept of angular magnification?

The moral of the lesson: the Newtonian lens formula xx′ = f² tells us the magnification, y′/y, in terms of the extrafocal distances measured from the focal points and the focal length of a magnifying device.

 

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.

3. Landsberg, G. S. (2000). Textbook of Elementary Physics, Volume 3 (A. Troitsky, Transl.). Honolulu, Hawaii: University Press of the Pacific.