An Elementary Treatise on Optics/Chapter 12

by Henry Coddington

CHAP. XII.

IMAGES PRODUCED BY REFRACTION.

119.Let $ABC$ (Fig. 132.) represent a straight line or rod sunk in water up to $B$. The rays of light proceeding from any point $C$ in the water will be refracted so as to proceed apparently from a point $C',$ which, by referring to Chap. vii., we find is higher than $C$ by one-fourth of its depth, (supposing $m$ to be 3/4 for water).

The consequence of this is, that the visible appearance of the rod is such as is represented by the broken line $ABC',$ the tangent of the apparent angle of inclination to the surface ($C'BD$) being three-fourths of that of the real angle $CBD$.

120.Let us now examine the images produced by curved refracting surfaces. It will be quite sufficient in a practical point of view to take the cases of the double-concave and convex-lenses.

In these we find that when a pencil of rays is incident either along the axis or in a direction little inclined to it, the focal distance is given by the equation

${\frac {1}{\Delta '}}={\frac {1}{F}}+{\frac {1}{\Delta }}.$

From which we may conclude, that if $\Delta$ be constant, $\Delta '$ will be so; that is, if the object be a portion of a spherical surface, (Fig. 133.) having for its centre that of the lens, the image will be a similar portion of a sphere, the radius of which will be $\Delta '$ as that of the object is $\Delta$.

121.It the object be a straight line, it will be found that the case is precisely analogous to that in Chap. vi., substituting the center of the lens for that of the reflecting surface; the image is as there a portion of a conic section, the latus rectum of which is the principal focal length of the lens. The curvature of the image is the same for all distances.

The reader will have no difficulty in collecting from this that the image of the Sun given by a lens is a small portion of a spherical surface, the middle of which is at the principal focus of the lens.

It will moreover be easily seen (Fig. 134 and 135.) that a double-concave lens gives an erect image, a double-convex generally an inverted one. I say generally, because when the object is nearer the lens than the principal focus, (Fig. 136.) the image is beyond it on the same side, and therefore erect.

With the exception of this last case, the image given by a convex-lens is a real one, that by a concave-lens is always imaginary.

Note. In all images produced by reflexion or refraction at spherical surfaces, there is in practice necessarily some little indistinctness arising from the aberration of the rays distant from the axis. In consequence of this, instead of single points or foci we find small circular spots with bright edges and centres which intersect and confuse each other as represented on an exaggerated scale in Fig. 137.