After our in-class exercise with ray-tracking, you already know how to do it. However, I’d like to add some extra comments, explaining in detail the mea- ning of the arrows we draw for the “object” and the “image” – what is exactly their role in the ray-tracing diagrams.

Below is a ray-tracing diagram for a converging lens – something you already know very well. But let’s take a closer look at the object and at the image, using a magnifying glass: There is a point source of light, and the image is also a point. The ray-tracing method enables one to find the point image of a point object formed by the lens. The left arrow is not a part of the object, and the right one is not a part of the image! Then, what are these arrows for?! Is it really necessarry draw them?

Wouldn’t it be OK to make ray-tracing diagrams just like this one? Well, such a diagram is “essentially correct”. It looks “somewhat silly”, doesn’t it? And it may be confusing.

The arrows show where exactly the point object and the point image are located. They add much clarity to the diagrams! Therefore, we should always draw them -- however, keep in mind that they are not themselves objects or images, just “helpful indicators”.

Point objects are interesting – but primarily for astronomers (stars are good examples). In most „real-life” situations, however, we deal with objects of finite size – e.g., like the “rod” pictured below. Can we use ray tracing for such objects?

Sure! – why not? Simply think of the “rod” as of a “chain” consisting of a large number of point sources, and then do ray tracing for each “point source”, one by one!

Such a ray tracing procedure, though, would not be very convenient if done on paper. The large number of rays drawn would make the plot pretty messy – look:

However, it is not necessary to do the ray tracing for all our “point sources”. It’s enough to do the tracing only for the object endpoints – and we will get the image’s endpoints, which is all we need. Of course, the rod needs not to at a position symmetric relative to the lens axis – one may shift it up or down, ray tracing performed for the two endpoints only always give us the right posi- tions of the image’s endpoints.

And, of course, dividing the object into many “point sources” was needed only to explain the underlying idea – having understood it, we don’t need to plot individual “point sources” any more. We can plot the rod “as it is”, and do the ray tracing only for its ends – and then just plot the “image rod” by drawing a line between the two endpoints we have obtained. So much about the ray tracing procedures for large objects! And now we switch to the next important topic – magnification.

First, let’s define the so-called “lateral magnification”: B’ A ho O hi B A’ xo xi

Quick quiz ( not written, verbal): Object far away from the lens (xo >> f ): Is the magnification ML a large number ( >>1 ), or a small number ( << 1 )? Can you think of a device that is a good example of such situation? 2. Object close to the lens ( xo only slightly larger than the focal length f ): Is the lateral magnification a large number, or a small one? (Hint: one such device is here, in this very classroom!). The symbol “>>” means “much larger”, and “<<“ means “much smaller”.

However, for us a more interesting and more important parameter is the so-called ANGULAR MAGNIFICATION First. let’s define what we call the ANGULAR SIZE of an object – the picture below explains what it is: The angular size (AS) of an object depends on how far it is from the eye. The closer is the object, the larger is its angular size. The AS of a dime viewed from the distance of 1 yard is about 30 minutes of arc. From 30 yards, it’s about a single minute of arc. Human eye cannot resolve details smaller that a few minutes of arc. Looking at a dime from 30 yards, you can probably recognize that it’s a coin – but you rather won’t be able to tell whether it’s an American coin, or a Canadian dime. For “seeing things better”, we always want to bring them closer to our eyes – i.e., we want to make their angular size bigger.

Angular magnification, not lateral magnification, is the one that really matters when we talk about instruments used for direct visual observations. Last time, we did ray-tracing for a simple two-lens microscope. from the plot, it is clear, that the image is indeed considerably magnified. But it still does not show that the angular magnification is big. In order to see that, we need to add the eye of the observer to the picture – it’s on the next slide.

The angular size of the object observed by an unaided eye is the angle between the lens’ axis and the red line in the picture. The angu- lar size of the image is the θ2 angle.

Angular magnification is particularly important in the case of telescopes – it is, instruments used for observing very distant objects. Talking about lateral magnification in the case of telescopes does not make much sense! Why? The reason is simple: because we usually don’t know how far the object is, and what are its dimensions. It’s only the angular magnification that matters. If you see a telescope in a store, with a label “ 60×”, it means that the angular size of the of the object’s image produced by this telescope is sixty times the angular size of the same object viewed by an unaided eye. For instance, the angular size of full Moon is about 30 minutes of arc; watched by this instrument it would be of the size of a vinyl LP record held in extended hand.