Now we’re ready to discuss lenses. Lenses will follow many of the same properties or the same logic
that we had when we were discussing mirrors. For a lens, again you can have something that is
concave or convex but there’s actually a different sort of terminology. Sometimes we call this
concave lens a diverging lens because as the light goes into it, it gets dispersed in many different
directions. Then we call the convex lens, the outward lens, the converging lens because as light
goes into it, it converges towards some point. We could think of a lens as being again, part of a circle
where instead of just one circle, we have a circle for each side of the lens. Each lens also has a focal point.
We’ll be considering thin lenses so that we don’t have to worry about the effects of the light
as it goes through the lens and bends with refraction each time. For a thin lens like this one,
again, we also have a focal point which we’ll discuss more in a moment. We also have a center
of the curvature, if we again drew for ourselves that circle that we already showed.
Then, we could follow the same logic for putting some object again just some arrows, so that we have
some orientation to talk about on one side of the lens and seeing where the corresponding image
would appear on the other side of the lens. The two rays we’ll draw from the arrow, in this case
are very similar to the ones from the mirror. We have one going right through the focal point.
That one, after hitting the lens will leave parallel to the lens. The other ray that we’ve drawn will be parallel
to the axis. Then on the other side of the lens, rather than reflecting, it will go through the lens
and pass through the focal point on the opposite side of the lens. The second ray after it’s gone
through the lens itself and passing the second focal point will meet up with the first ray that we
drew which went through the focal point and then left parallel. Then we have the new image
distance that we perceived for this particular lens. Again, this follows the same logic.
If I were on the other side of this lens where I would see the two rays coming out from this whole procedure,
I would think that those two rays were coming from some point that they weren’t actually coming
from because I wouldn’t be able to process that they might have bent on their way to my eyes.
So, I would perceive this image, this image that’s on the right of the lens. For a diverging lens,
we have something slightly different. We first draw a parallel ray to the axis and then say that
that ray diverges directly away from the near focal point. For the other ray, we would draw
towards the far focal point. It would refract to be parallel to the axis. You can see in this case,
we’re always following the same sort of logic. We have a first ray and a second ray, one of which
is trying to be parallel to the axis and the other one is trying to go towards a focus.
We can always see how the focus and the parallel nature of the rays is playing back and forth to find
where the rays will go. Again, we’ll do the same logic for finding the image, where the image would be
and what the image would look like. We take the final point of those two rays. We ask ourselves
the one question that we keep bringing up which is where would my eye perceive these rays
to have come from? We just trace those two rays back until they finally meet. In this case,
we can see that the image which is where the rays will meet is going to be found on the near side
of this lens. We would perceive a smaller image of our object. As a summary for what we’ve talked
about with lenses, if we have parallel rays coming in from an object that is sort of at an infinite distance,
if you will, the rays will all converge to some focal point. If we have an object that is near
the lens but still outside of the focal point, we end up with an inverted real object. If our object
is on the focal point, we’ll have the same problem of each beam leaving at a parallel, in a parallel
path so they’ll never meet. Then finally, for an object inside the focal point, we could trace our
rays in the same way that we’ve been doing, one going straight towards the lens, parallel, and
then going through the final focal point, the other one going directly away from the near focal
point and leaving parallel. Then, we could trace these two rays back. Again, we would find
an upright, virtual image. For diverging lens, we have the same sort of idea that we had for
a convex mirror which is that the rays impinging from an object for a diverging lens are always
going to cause the same sort of thing to occur rather than having many options. For any object
anywhere near a diverging lens, we can draw one ray parallel which will leave away from the focal
point, one directly towards the opposite focal point which will leave parallel. Then all of these cases
will always end up with an upright, virtual image. For the locations of the objects that we’ve discussed,
we’re going to go back to this little letter o and try to figure out some things about the locations.
Again, for lenses just as for mirrors, we can define a location o for our object, a location i
for our image, and then a location for our focus. Again, we’ll call that f. We have the exact same equation
that we had for mirrors as we will have for lenses. The thing to worry about or the thing to be concerned
about here is that the conventions can be a little bit different. We have the same lens equation
for how the object, image, and focal distances are related but our conventions are slightly different
because of course, for lenses, the rays can go through the lens rather than bouncing off as they did
for a mirror. The conventions for a lens are that the focal point is positive for a converging lens
and negative for a diverging lens. The object distance is again always going to be positive.
This is sort of by convention. If we try to put the object on the other side of the lens, we could always
just turn ourselves around and look at it from the other point of view. We’ll always sort of consider
the object distance itself to be a positive number. Finally, the image distance is going to be a positive
number for real images but a negative number for virtual images no matter where those are.
This concludes our lecture on mirrors and lenses. There’s a lot of material in this lecture with regard
to the distances of the image, and the object, and the focus as well as how to treat both mirrors
and lenses. So, I definitely recommend that you would go back over it and maybe try some practice
problems. As you sort of let it sit for a while and maybe go over it a few more times, hopefully
it becomes more comfortable because again, this procedure that we introduced for this parallel
rays and then the rays going to the focus, these procedures are just used over and over again.
Once you sort of get an intuition for when should I use parallel rays, when should I use rays towards
the focus, when will they leave and go away from the focus, for example, how can I use my lens
and mirror equation, these things will all become more and more intuitive with even a few
practice problems. I highly recommend you do that. We have one more section on optics
which is coming up next. Thanks for listening.