Stem-Master
Physics / Chapter 34: Images / Section 34-8

Three Proofs — Question 8

Section 34-8 · Physics pages 5706–5707

Question 8 Tracing one parallel ray proves the power of separated thin lenses Two thin lenses in air have focal lengths f1 and f2 and separation d. An incoming parallel ray reaches lens 1 at height h. A lens changes slope by minus height divided by focal length. Define the equivalent focal length F through outgoing slope u2 = −h/F. d first second Schematic; labels, not drawing lengths, specify the data. (a) Trace the ray height and slope through both lenses and derive 1/F. (b) Derive the back focal distance measured from lens 2 and identify the afocal condition.
Show solutionHide solution
Question 8 - Solution Tracing one parallel ray proves the power of separated thin lenses d first second (a) Method and calculation u1 = −h/f1, y2 = h + du1 = h(1 −d/f1). u2 = u1 −y2/f2 = −h[1/f1 + 1/f2 −d/(f1f2)]. (b) Interpretation and further calculation 1/F = 1/f1 + 1/f2 −d/(f1f2), qback = −y2/u2 = (1 −d/f1)F . 1/F = 0 ⇒d = f1 + f2 for finite nonzero focal lengths. Physical check The back focal distance is finite only when the outgoing slope is nonzero; an afocal pair sends the parallel input to parallel output.

Question 8

Tracing one parallel ray proves the power of separated thin lenses

Two thin lenses in air have focal lengths f1 and f2 and separation d. An incoming parallel ray reaches lens 1 at height h. A lens changes slope by minus height divided by focal length. Define the equivalent focal length F through outgoing slope u2=−h/Fu_2=-h/F.

Diagram shown in the original SVG worksheet.

  1. Trace the ray height and slope through both lenses and derive 1/F.

  2. Derive the back focal distance measured from lens 2 and identify the afocal condition.

Question 8 - Solution

Tracing one parallel ray proves the power of separated thin lenses

Diagram shown in the original SVG worksheet.

(a) Method and calculation

u1=−h/f1,y2=h+du1=h(1−d/f1).u_1=-h/f_1,\quad y_2=h+du_1=h(1-d/f_1).u2=u1−y2/f2=−h[1/f1+1/f2−d/(f1f2)].u_2=u_1-y_2/f_2=-h[1/f_1+1/f_2-d/(f_1f_2)].

(b) Interpretation and further calculation

1/F=1/f1+1/f2−d/(f1f2),qback=−y2/u2=(1−d/f1)F.\boxed{1/F=1/f_1+1/f_2-d/(f_1f_2),\quad q_{\mathrm{back}}=-y_2/u_2=(1-d/f_1)F}.1/F=0⇒d=f1+f2for finite nonzero focal lengths.1/F=0\Rightarrow\boxed{d=f_1+f_2}\quad\text{for finite nonzero focal lengths}.

Physical check

The back focal distance is finite only when the outgoing slope is nonzero; an afocal pair sends the parallel input to parallel output.

← Section questionsChapter contents