Assume two thin converging lenses with focal length f1 and f2, respectively, are very close to each other, so we can neglect the distance between them. They focus the light of a distant star at a focal point a distance feff from the lenses.  Find feff in terms of f1 and f2.

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Reflection When a wave hits a smooth object that is larger than the wave itself, depending on the media, the wave may bounce in another direction.

Do you have a magnifying glass?  If not, do you have access to some prescription eyeglasses?  Make some observation with a single lens.

A magnifying glass is a converging lens of focal length 15 cm.  At what distance from a postage stamp should you hold this lens to get a magnification of +2?

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A linear pair is a pair of adjacent angles formed when two lines intersect.

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The lens and mirror equations are valid for paraxial rays.  If all the light rays involved in image formation make small angles with the optic axis, so that sinθ ≈ tanθ ≈ θ, then near perfect images form at the locations predicted by these equations.  For off-axis or skew rays, the image of a point becomes blurry and we observe aberrations.

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Virtual image produced by converging lens <--> M > 1, (image is larger than object) Virtual image produced by diverging lens <--> M < 1, (image is smaller than object)

A converging lens of focal length 4 cm is used as a magnifier.  If an object is placed 3 cm from the lens, what is the magnification?

A contact lens is made of plastic with an index of refraction of 1.5.  The lens has an outer radius of curvature of 2 cm and an inner radius of curvature of 2.5 cm. What is the focal length of the lens?

A thin lens has a focal length of 25 cm.  Locate the image when the object is placed(a) 26 cm(b) 24 cmin front of the lens.  Describe the image in each case.

A plane through a focus perpendicular to the optic axis is called a focal plane.  Parallel rays converge in the focal plane or diverge from the focal plane, independent of the angle they make with the optic axis.

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The lens equation and the mirror equation are written as 1/xo + 1/xi = 1/f.But the sign conventions for xo, xi, and f are different for lenses and mirrors.

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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.

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(b) xi = 25*24/(24 - 25) = -600. The image a virtual image 600 cm in front of the lens.Its magnification is M = 600/24 = 25. The image is upright and enlarged.

A single spherical interface between two transparent media can lead to image formation.  But viewing these images is not always possible.  Consider an air-glass interface.  When the object is on the air side, the image can only be viewed from inside the glass.  Giving the glass a finite thickness and adding a second spherical interface between glass and air will produce an image that can be viewed by someone in air.  A piece of glass of finite thickness with two spherical boundaries is a lens.  If the thickness of the lens is much smaller than the diameter of the lens, we call it a thin lens. Thin lenses can be converging or diverging.  Converging lenses are thicker in the middle than near the edges, and diverging lenses are thicker near the edges than in the middle.  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.  Rays parallel to the optic axis passing through a thin converging lens come together at the focus f on the opposite side of the lens, and rays parallel to the optic axis passing through a diverging lens diverge as if they were leaving the focus on the incident side on a straight line path.

Applying the lens equation to thin lenses, we can safely assume the light bends at the center of the lens, and neglect the thickness of the lens.  For thick lenses we have to be more careful.One must calculate the location of principal planes, from which to measure distances to the object, the image, and the focal points.

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The focal length of a thin lens is related to the radii of curvature of its two surfaces. 1/f = (n2 - n1)(1/R1 - 1/R2).

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.