Focal Length of a Zoom Lens. Figure P34.101 shows a simple

QUESTION:

Focal Length of a Zoom Lens. Figure P34.113 shows a simple version of a zoom lens. The converging lens has focal length \(f_{1}\), and the diverging lens has focal length \(f_{2=-}\left|f_{2}\right|\). The two lenses are separated by a variable distance  that is always less than . Also, the magnitude of the focal length of the diverging lens satisfies the inequality \(\left|f_{2}\right|>\left(f_{1}-d\right)\). To determine the effective focal length of the combination lens, consider a bundle of parallel rays of radius  entering the converging lens. (a) Show that the radius of the ray bundle decreases to \(r_{0}^{1}=r_{0}\left(f_{1}-d\right) / f_{1}\) at the point that it enters the diverging lens. (b) Show that the final image  is formed a distance \(s_{2}^{1}=\left|f_{2}\right|\left(f_{1}-d\right)\left(f_{2} \mid-f_{1}+d\right)\)
to the right of the diverging lens. (c) If the rays that emerge from the diverging lens and reach the final image point are extended backward to the left of the diverging lens, they will eventually expand to the original radius  at some point  The distance from the final image  to the point  is the effective focal length  of the lens combination; if the combination were replaced by a single lens of focal length  placed at , parallel rays would still be brought to a focus at . Show that the effective focal length is given by \(f=f_{1}\left|f_{2}\right|\left(\left|f_{2}\right|-f_{1}+d\right)\) (d) If \(f_{1}=12.0 \mathrm{~cm}\),
\(f_{2}=-18.0 \mathrm{~cm}\), and the separation  is adjustable between 0 and , find the maximum and minimum focal lengths of the combination. What value of  gives

Equation transcription:

Text transcription:

F_{1}

f_{2=-}\left|f_{2}\right|

\left|f_{2}\right|>\left(f_{1}-d\right)

r_{0}^{1}=r_{0}\left(f_{1}-d\right) / f_{1}

s_{2}^{1}=\left|f_{2}\right|\left(f_{1}-d\right)\left(f_{2} \mid-f_{1}+d\right)

f_{2}=-18.0 \mathrm{~cm}

f=f_{1}\left|f_{2}\right|\left(\left|f_{2}\right|-f_{1}+d\right)

f_{1}=12.0 \mathrm{~cm}