Thin lenses

•  Light refracts through the surface of a lens
•  The curvature of the lens causes light rays to converge or diverge
•  Light travels in straight lines
•  An eye sees by focusing a bundle of diverging rays
• A converging lens uses refraction to focus the rays

•  Rays from the tip of the arrow converge below the optical axis
•  Rays from the bottom of the arrow converge above the optical axis

Snell's law determines how sharply the rays bend

•  A double concave lens causes parallel rays to diverge
•  The focal length is the distance from the lens plane to the focal point
•  The focal length is a property of the lens
•  Any initially parallel ray will refract through the lens and pass through the focal point
•  A ray passing through the center of the lens is a straight line

Any ray passing through the near focal point emerges as a parallel ray

•  A real image is created through a converging lens
•  Notice the "special rays" called the principal rays
•  An incident parallel ray passes through the far focal point
•  A ray passing through the center of the lens doesn't bend
•  A ray passing through the near focal point emerges parallel
•  Note the real image is inverted

Practice problem:

The focal length of a converging lens is:

A. the distance from the object where an image is formed

B. the distance at which an object must be placed to form an image

C. the distance from the lens plane at which parallel light rays are focused

D. the distance from the front surface of the lens to its back surface

E. the point at which light is focused inside the lens

• We can use similar triangles to calculate the image distance, given the object distance and the object and image heights.
• Magnification is the negative ratio of the image distance over the object distance.
• Note that the image distance is measured as the distance to the right of the lens, for light propagating from the left.
• In this case the image distance is positive, making the magnification a negative number.
• A negative magnification implies an inverted image.
• In this case, the image is a real image.

A real image (left) can be projected on a screen. The light rays emanate from the object, go through the lens, and project onto the screen. Real images are always inverted.

A virtual image (right) cannot be projected on a screen. Notice you can see this candle image when looking into the lens itself. Virtual images are always upright.

The thin lens equation relates the focal length, f, to the object distance, s, and the image distance, s'.

Practice problem:

A 4.0 cm high object is 150 cm from the nearest focal point a 50 cm lens of a camera as shown.

How far should the film be placed behind the lens to record a focused image?

What is the height of the image on the film?

Draw the ray diagram to show the image.

Note that the image is inverted. What is the magnification?

Notice that your object is more than two focal lengths from the lens.

• Does your answer change qualitatively if the object is placed between one and two focal lengths from the lens? How?

• What happens if the object is placed two focal lengths before the lens?

• Does your answer change qualitatively if the object is placed between the focal point and the lens? How?

• What happens if the object is placed right at the focal point?

Practice problem:

Which of the following statements is true, concerning a double convex lens?

A. A real image is formed if s < f, a virtual image is formed for s > f

B. A virtual image is formed if s < f, a real image is formed for s > f

C. A real image is always formed

D. A virtual image is always formed

Diverging lenses

•  A double concave lens causes rays to diverge
•  Any initially parallel ray will diverge through the lens along a line that passes through the near focal point

• Any ray directed toward the far focal point emerges as a parallel ray

• Any ray passing through the center of the lens is unbent

Practice problem:

A diverging lens with a focal length of 50 cm is placed 100 cm from an object.

• Where is the image?

• What is the magnification?

Draw the principle rays

• One parallel before the lens
• One through the lens center
• Trace the rays backward to find where they converge.

Ray diagrams - summary

The second lens in this optical instrument

A. Causes the light rays to focus closer than they would with the first lens acting alone

B. Causes the light rays to focus farther away than they would with the first lens acting alone

C. Inverts the image but does not change where the light rays focus

## Spherical mirrors

Analysis for concave and convex spherical mirrors is similar to that for concave and convex lenses.

Similarly to our lens analysis, for spherical mirrors we draw principle rays from the object to the mirror and then to the image. The diagram above shows the principle rays for a concave mirror. A real, inverted image is formed on the same side of the mirror as the object.

We use the same equations we used for lenses, with a sign convention for mirrors. For a concave mirror, the focal point in front of the mirror is positive. The image distance s' is also positive.

This diagram shows the principle rays for a convex mirror. Notice that here, a virtual, upright image is formed inside the mirror.

The sign convention for a convex mirror is that the focal point beyond the mirror plane is defined to be negative and the image distance is also beyond the mirror plane and is defined to be negative.