7.4.1: Illustrations

Illustration 1: Lenses and the Thin-Lens Approximation

Shown is an optics bench that allows you to add various optical elements (lens, mirror, and aperture) and light sources (beam, object, point source) and see their effect. Elements and sources can be added to the optics bench by clicking on the appropriate button and then clicking inside the animation at the desired location. Moving the mouse around in the animation gives you the position of the mouse, while a click-drag will allow you to measure angle (position is given in centimeters and angle is given in degrees).

Add a lens to the optics bench by clicking on the "Lens" button and then clicking inside the animation to place the lens. Adjust the focal point of the lens by dragging on the round hotspots. Notice that you can make the lens either converging or diverging. Make the lens converging with a focal length of \(1\text{ cm}\) and place it in the middle of the animation (\(x = 2.5\text{ cm}\)). Now add a source of light by clicking on the "Object" button and then clicking inside the animation. Place the object at \(x = 0.1\text{ cm}\) and give it a height of \(0.5\text{ cm}\). You can add other sources of light later.

Notice the rays emanating from the object, their refraction (bending), and the resulting image. Click the head of the arrow (the object) and move it around. First note that there are three rays that emanate from the head of the arrow. One ray comes off parallel to the principal axis (the yellow centerline) and is bent at the vertical centerline of the lens and then travels through the focal point on the far side of the lens, one ray comes off at an angle to hit the vertical centerline of the lens on the principal axis and is unrefracted, and one ray passes through the principal axis at the near focal point of the lens and is bent at the vertical centerline of the lens and comes off parallel to the principal axis.

Do the rays always behave as you expect them to? Probably not. As you drag the head of the source and change its height and position, what do you notice about the rays when they refract through the lens? The rays refract from the vertical centerline of the lens. If you click on the lens, you will see this line in blue. This animation uses what is called the thin-lens approximation. This approximation assumes that the lens is thin in comparison to its radius of curvature. In fact, in the thin-lens approximation, we take the thickness of the lens to be zero (this is why the refraction takes place at the vertical centerline of the lens). In a real lens, the rays from the object would form a real image to the right of the lens, but also a virtual image behind the object on the left of the lens.

The optics bench allows you to try many different configurations to see how light will interact with a lens. Take some time to play with the animation. You may also find it helpful to refer back to this Illustration as you develop your understanding of optics. A brief description of the three sources is given below.

• The "Beam" button adds a beam of parallel light rays. The angle of the light rays can be changed by dragging the hotspot after clicking on the beam.
• The "Object" button adds an arrow as an object. A ray diagram is drawn for the object if an optical element is present.
• The "Source" button adds a point source of light. The spread of the light rays can be adjusted by dragging the hotspot after clicking on the source.

Problem authored by Mario Belloni and Melissa Dancy.

Illustration 2: Image from a Diverging Lens

An object (the arrow) is in front of a diverging lens. You can drag the lens around, but the object is fixed. Moving the mouse around in the animation gives you the position of the mouse, while a click-drag will allow you to measure angle (position is given in centimeters and angle is given in degrees). Notice the rays emanating from the object, their refraction (bending), and the resulting image. In this case the image is virtual. Restart.

To determine where the image would be located and that it is indeed virtual, place the lens at \(x = 2\text{ cm}\). Note that there are three rays that emanate from the head of the arrow.

• One ray comes off parallel to the principal axis (the yellow centerline), is bent at the vertical centerline of the lens, and then travels to the right on a line that, if continued backwards to the left, would pass through the focal point on the near side of the lens.
• One ray comes off at an angle to hit the vertical centerline of the lens on the principal axis and is unrefracted.
• One ray, which if continued to the right would pass through the focal point on the far side of the lens, is bent at the vertical centerline of the lens and comes off parallel to the principal axis.

Given these statements, can you determine the focal length of the lens? On one (or both) of the lines that if continued would pass through a focal point, click-drag the center circle of the compass tool into position over one of those lines. Now drag the green line until it is parallel to the ray and passes through the principal axis (the yellow centerline) so that the animation looks like the following image.

Figure \(\PageIndex{1}\)

If the lens is at \(x = 2\text{ cm}\), you should get the position at which the green line crosses the principal axis to be \(x = 1\text{ cm}\) or \(x = 3\text{ cm}\), depending on which ray you choose. In either case, since the lens is at \(x = 2\text{ cm}\), the focal length of the lens is \(1\text{ cm}\).

How do we determine where the image will be located? Notice that the three rays diverge. The dotted lines continue the three rays on the right of the lens backward on the left of the lens to a point where all three dotted lines converge. This is the location of the image. It is a virtual image: A screen placed at the image location would not be illuminated by the image of the object.

Illustration authored by Melissa Dancy and Mario Belloni.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.