This lenses can be converging or diverging lenses. A thin lens has two focal points, located on the optical axis, a distance f from the center of the lens on either side of the lens. Parallel rays passing through a thin converging lens come together at the focus f on the opposite side of the lens, and parallel rays are passing through a diverging lens diverge as if they were leaving the focus on the incident side on a straight line path.
Converging lens
Diverging lens
Thin lenses can form real and virtual image.
Let xo denote the perpendicular distance of the object from the centerline of the lens and let xo be positive. Let xi denote the perpendicular distance of the image from the centerline of the lens. Then xi can be found from the lens equation
1/xo + 1/xi = 1/f,
provided we use the following sign convention.
 | xo is positive. |
| xi is positive if xo and xi are on the opposite sides of the lens. |
| xi is negative if xo and xi are on the same side of the lens. |
| f is positive for a converging lens. |
| f is negative for a diverging lens. |
| The magnification is M = -xi/xo. If M is negative, the image is inverted. |
We can find the position and size of the image geometrically. Only two rays must be drawn.
| Draw the optical axis and the centerline of the lens. | ||
| Mark the position of the foci. | ||
| Draw the object in front of the lens. Draw an incident
ray parallel to the optical axis from a point on the object to the centerline of the lens,
and a refracted ray from the centerline through f.
| ||
| Draw a second incident ray through f, and a refracted ray
parallel to the optical axis.
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| The intersection of the two rays marks the position of the image. | ||
| To check the accuracy of your drawing, draw a third ray through the center of the lens. This ray is not bent. It should pass through the intersection of the other rays that you have drawn. |
Diverging lenses only form virtual images. Converging lenses form real inverted images if xo > f and virtual upright images if xo < f.
The focal length of a lens is related to the radii of curvature of its two surfaces
1/f = (n2-n1)(1/R1-1/R2)
| R is positive, if xo and the center curvature are on the opposite sides of the lens. |
| R is negative, if xo and the center curvature are on the same side of the lens. |
| R1 is the radius of the surface closest to xo. |
Many optical instruments have more than one optical element. A combination of mirrors and lenses can be analyzed by treating the image of the first element as the object of the second element, and so on. If the image of the first element falls behind the second element we can still use the lens or mirror equation, but we then must use a negative object distance xo.
Problems:
| A contact lens is made of plastic with an index of refraction of
1.5. The lens has an outer radius of curvature of 2cm and an inner
radius of curvature of 2.5cm. What is the focal length of the lens?
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| A thin lens has a focal length of 25 cm. Locate the image when the
object is placed (a) 26cm (b) 24 cm in front of the lens. Describe the image in each case.
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| A magnifying glass is a converging lens of focal length 15cm. At what
distance from a postage stamp should you hold this lens to get a
magnification of +2?
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Exercise (You can earn up to 5 points extra credit by completing this exercise.)
Links to other Web material:
| Image formation by lenses |
| The eye |
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