A converging lens and a diverging lens, separated by a...

A converging lens and a diverging lens, separated by a distance of $30.0 \mathrm{~cm}$, are used in combination. The converging lens has a focal length of $15.0 \mathrm{~cm}$. The diverging lens is of unknown focal length. An object is placed $20.0 \mathrm{~cm}$ in front of the converging lens; the final image is virtual and is formed $12.0 \mathrm{~cm}$ before the diverging lens. What is the focal length of the diverging lens?

Key Concepts

Effect of Lens Separation

The separation between lenses in a combined optical system plays an important role in determining the effective object distances for subsequent lenses. The distance between the lenses must be factored into the positioning of the intermediate image from the first lens to correctly compute the object distance for the second lens, influencing the final image properties.

Thin Lens Equation

The thin lens equation, given by 1/f = 1/do + 1/di, is fundamental in geometrical optics for relating the focal length of a lens (f) to the distances of the object (do) and the image (di) from the lens. This relation is used to determine unknown parameters in lens systems under the paraxial approximation.

Lens Sign Conventions

In optics, sign conventions are used to indicate the positions and types of objects and images relative to a lens. Positive and negative signs in the lens formula denote real versus virtual images and objects, and they are essential for correctly applying the thin lens equation when working with converging and diverging lenses.

Sequential Imaging with Combined Lenses

When multiple lenses are used in a system, the image formed by the first lens often serves as the object for the second lens. Analyzing such a system requires a careful step-by-step application of the lens equation to each lens, accounting for the distance between the lenses, so that the overall behavior of the system can be correctly determined.

Virtual Image Formation

Virtual images occur when the outgoing rays from a lens do not converge but rather appear to diverge from a common point. In optical systems, a virtual image is formed when the calculated image distance is negative according to the sign conventions. Understanding virtual image formation is crucial in scenarios where the final image is not captured on a screen but is observed by the eye.

Transcript

00:01 So this problem here has kind of a complicated situation where you have a converging lens and you get a diverging lens over here.

00:10 And this one has a focal length.

00:12 It's given a 15 centimeters.

00:16 And then 20 centimeters in front of it is the object.

00:21 And then what happens is that this object is going to make an image somewhere.

00:28 And that image, i'm going to call it d .i.

00:31 I is going to become the object for the second lens.

00:40 Now the deal with the second lens is that it has an unknown focal length, unknown.

00:48 But we are told that 12 centimeters in front of this lens is where we end up with our image, the i2.

00:59 All right.

01:01 So this is the deal here.

01:03 We have to figure out what is the focal length of this lens here.

01:07 So i'm going to start off by dealing with the problem of the first lens to find out where is d .i .1.

01:14 Once i know what d .i .1 is, and i know where it is on here, i'll be able to go back and solve for what the focal length of the second one is.

01:23 So let's use the lens equation.

01:27 1 over 15 centimeters is equal to 1 over 20 plus 1.

01:35 Over d .i .1.

01:38 All right, 60ths, that's four 60ths is equal to three 60ths plus one 60th.

01:47 So, d .i .1 equals 60 centimeters, and that's in this direction over here.

01:58 Now, we are told that these lenses are 30 centimeters apart.

02:05 And if this is the image of the first one is 60 to the right, we are now understand that this will also be 30 centimeters right here.

02:17 So in other words, d -i -1 is not equal to d -o -2 because there's a distance between the lenses.

02:24 So now we have the following information.

02:30 We have the lens...

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