Chapter 34, Problem 34.101P

Focal Length of a Zoom Lens. Figure P34.101 shows a simple version of a zoom lens. The converging lens has focal length f₁ and the diverging lens has focal length f₂ = −|f₂|. The two lenses are separated by a variable distance d that is always less than f₁ Also, the magnitude of the focal length of the diverging parallel rays of radius r₀ entering the converging lens, (a) Show that the radius of the ray bundle decreases to ${r^{\prime }}_0=r_0(f_1-d)/f_1$ at the point that it enters the diverging lens. (b) Show that the final image I′ is formed a distance ${s^{\prime }}_2\text{=}\left|f_2\right|\left(f_1-d\right)/\left(\left|f_2\right|-f_1+d\right)$ 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 r₀ at some point Q. The distance from the final image I′ to the point Q is the effective focal length f of the lens combination; if the combination were replaced by a single lens of focal length f placed at Q, parallel rays would still be brought to a focus at I′. Show that the effective focal length is given by f = f₁|f₂|/(|f₂| − f₁ + d). (d) If f₁ = 12.0cm, f₂ = − 18.0 cm, and the separation d is adjustable between 0 and 4.0 cm, find the maximum and minimum focal lengths of the combination. What value of d gives f = 30.0 cm?

Figure P34.101