The figure below shows a candle with a flame near the left side of the picture.

• Three rays are depicted emanating from the tip of the flame: a ray parallel to the central axis of the lens, a ray passing through the center of the lens, and a ray striking the lower part of the lens at such an angle that the transmitted ray ends up parallel to the axis of the lens.
• The third of these rays will pass through a point located at the focal distance on this side of the lens.
• The distance of the candle from the lens is called the object distance, and is indicated by the o in the figure.

In the next figure we see the same rays after they pass through the lens.

• The top ray, being parallel to the axis of the lens, will be deflected through the focal point as indicated.
• The middle ray, striking the middle of the lens, will pass through without being significantly deflected.
• The bottom ray, as stated earlier, ends up parallel to the central axis of lens.
• All three rays converge at a point.

Using straightforward geometry, we can show that the image distance i, the object distance o and the focal distance f must satisfy the equation 1/f = 1/o + 1/i indicated in the figure below.

Video Clip #10

In the preceding figure the image was inverted, upside down from the object.

• If we were to move a screen to a point between the lens and the focal point, we could get a sort of an image, and it would be upright, and very blurry.

Video Clip #11

We see that has an object moves closer to a lens, the undeflected ray through the center and the ray parallel to the axis of the lens which is deflected through the focal point become more and more nearly parallel.

• The result is that the image forms further and further beyond the lens, and gets larger and larger.
• When the object reaches the point where its distance from the lens is equal to the focal distance, the rays on the other side of the lens will be parallel and will never converge.
• So we see that as the object distance approaches the focal distance, the image distance approaches infinity, as does the image size.

For a convex lens, parallel rays are deflected away from the axis of the lens.

• It follows that the paths of the rays meet on the same side of the lens as the object.
• The image formed is virtual, since the converging rays do not even exist at the point of the image.
• From the geometry of the situation is evident that the virtual image is also upright.
• Looking through the lens, the object appears smaller and closer than it actually is.

The focal point of a convex lens is the point where the paths of incoming parallel rays meet.

• As observe before, the rays do not really exist at this point and images formed will be virtual.
• In this case we say that the focal length of the lens is negative, and we continue to use the relationship 1/f = 1/o + 1/i.

We see how the negative value of f implies that, for a positive object distance (and object distance is always positive), the image distance i must be negative--otherwise one side of the equation would be negative in the other positive.

• Thus the equation also predicts that the image and the focus will be on the 'wrong side' of the lens.

Video Clip #13

The figure below depicts the two principal rays used to determine the position and size of the virtual image formed by a convex lens.

• The first ray is parallel to the axis of the lens and the second is the undeflected ray that passes through the center of the lens.

The figure below depicts a thin lens formed from a part of a circular or spherical object.

• The focal length, as we have seen, depends on the index of refraction in the radius of the circle or sphere.

Using the 3-inch and 4-inch water lenses, Josh obtained focal distances of about 12 cm and 15 cm respectively.

• For the 3-inch water lens it was therefore predicted that and image of an object 350 cm in front of the lens would be 12.5 cm behind the lens.
• By turning off the lights and illuminating an object at this distance, we did in fact obtain a very good image by placing a screen 12.5 cm behind the lens.

The image size for this situation is predicted to be 12.5 / 350 times the size of the object.

• This predicts an image approximately 2 inches high for a person 5' 6" tall.
• This is also consistent with our observations.

Images can be formed by more than one lens.

• When a combination of lenses is used, the first lens forms and image which then becomes the object for the second lens, as depicted below.
• We see that for the original object we have used to principal rays t2o locate the image.
• Then using this image, we used two principal rays through the second lens to locate the final image.