Question 33.6: Measuring f for a diverging lens. To measure the focal lengt......
Measuring f for a diverging lens. To measure the focal length of a diverging lens, a converging lens is placed in contact with it, as shown in Fig. 33-15. The Sun’s rays are focused by this combination at a point 28.5 \mathrm{~cm}, behind the lenses as shown. If the converging lens has a focal length f_{\mathrm{C}} of 16.0 \mathrm{~cm}, what is the focal length f_{\mathrm{D}} of the diverging lens? Assume both lenses are thin and the space between them is negligible.
APPROACH The image distance for the first lens equals its focal length (16.0 \mathrm{~cm}) since the object distance is infinity (\infty). The position of this image, even though it is never actually formed, acts as the object for the second (diverging) lens. We apply the thin lens equation to the diverging lens to find its focal length, given that the final image is at d_{\mathrm{i}}=28.5 \mathrm{~cm}.
If the diverging lens was absent, the converging lens would form the image at its focal point-that is, at a distance f_{\mathrm{C}}=16.0 \mathrm{~cm} behind it (dashed lines in Fig. 33-15). When the diverging lens is placed next to the converging lens, we treat the image formed by the first lens as the object for the second lens. Since this object lies to the right of the diverging lens, this is a situation where d_{\mathrm{o}} is negative (see the sign conventions, page 871). Thus, for the diverging lens, the object is virtual and d_{\mathrm{o}}=-16.0 \mathrm{~cm}. The diverging lens forms the image of this virtual object at a distance d_{\mathrm{i}}=28.5 \mathrm{~cm} away (given). Thus,
\frac{1}{f_{\mathrm{D}}}=\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{-16.0 \mathrm{~cm}}+\frac{1}{28.5 \mathrm{~cm}}=-0.0274 \mathrm{~cm}^{-1}
We take the reciprocal to find f_{\mathrm{D}}=-1 /\left(0.0274 \mathrm{~cm}^{-1}\right)=-36.5 \mathrm{~cm}.
NOTE If this technique is to work, the converging lens must be “stronger” than the diverging lens-that is, it must have a focal length whose magnitude is less than that of the diverging lens. (Rays from the Sun are focused 28.5 \mathrm{~cm} behind the combination, so the focal length of the total combination is f_{\mathrm{T}}=28.5 \mathrm{~cm}.)