Virtual Objects

We saw in the last post that convex lenses are used to correct hyperopic eyes.    We see in the figure above we show the image of a distant tree a convex spectacle lens would produce if the eye were not interposed.  This may be said to be a virtual object for the lens of the eye and we can analyse the system in terms of ray diagrams by splitting it into two parts.  First, the spectacle lens gives rise to  a virtual object for  the convex lens of the eye and second the action of the eye lens on this object gives rise to the final image on the retina.

In the figure below we represent a converging lens upon which falls a system of rays which, in the absence of the lens, would form an image, at O on the optical axis, and at which we take the virtual object in the form of this image.  We draw a ray, passing through the front focus F$_1$ and falling on the lens at P.  If it were undeviated it would meet the virtual object at S but in actuality emerges from

the lens parallel to the optical axis.  We draw a second ray QR which again would pass through S in the absence of the lens but in fact passes through the second focus F$_2$.  We take $f$ as the focal length of the lens, $x$ as the distance of the object from the first focus and $x'$ the distance of the image from the second.  Referring to the diagram we note that $\triangle$F$_1$CP and $\triangle$F$_1$SO are similar and hence $h/h'=x/f$.  Furthermore, $\triangle$F$_2$CR and  $\triangle$F$_2$IJ are similar and hence $h/h'=f/x'$.  We arrive at Newton's formula
\[ f^2=x'x\]
The lens formula is readily obtained from this, in a manner similar to that detailed in the post on virtual images. In fact, the two situations are the same geometrically, the directions of rays being reversed while object and image are interchanged.  This is also the case for the diverging lens with virtual object. The ray diagram is shown below.  We shall not discuss it further as Newton's formula is obtained in exactly the same way as before.

Virtual objects were included in the summary of cases of the lens formula in the post for the 21$^\mathrm{st}$ of November.