Thin Lens Equation and Magnification

• The thin lens equation relates object distance (u), image distance (v), and focal length (f): 1/f = 1/u + 1/v
• This equation assumes a thin lens, where thickness is negligible compared to u, v, and f.
• Sign conventions: u is positive for real objects, v is positive for real images, f is positive for converging lenses.

Focal Length Determination

• Focal length (f) is the distance from the lens where parallel rays converge (converging lenses) or appear to diverge from (diverging lenses).
• For converging lenses, focusing sunlight onto a point approximates the focal length.
• Diverging lenses require combinations of lenses or virtual image methods to determine f.
• Lensmaker's equation: 1/f = (n-1)(1/R1 - 1/R2), where n is the refractive index, and R1 and R2 are the radii of curvature.

Lens Magnification

• Magnification (M) is the ratio of image height (hi) to object height (ho): M = hi/ho.
• Magnification relates to object and image distances: M = -v/u; the negative sign indicates image inversion for real images.
• M > 1: enlarged image, M < 1: reduced image.
• Positive magnification: upright image, negative magnification: inverted image.

Applications in Optical Instruments

• Lenses are fundamental to cameras, telescopes, microscopes, and eyeglasses.
• In cameras, lenses focus light onto a sensor/film; focal length determines field of view and magnification.
• Telescopes use lenses (or mirrors) to collect and focus light from distant objects, magnifying images.
• Microscopes use multiple lenses to magnify small objects.
• Eyeglasses use lenses to correct vision by properly focusing light onto the retina.

Object-Image Distance Relationship

• The thin lens equation shows an inverse relationship between object and image distances given a fixed focal length.
• As object distance (u) decreases, image distance (v) increases, and vice versa.
• For converging lenses, when u = ∞, the image forms at the focal point (v = f).
• When u = f, the image forms at infinity (v = ∞).
• The minimum distance between the object and its real image for a converging lens is 4f, occurring when u = 2f and v = 2f.

Converging vs Diverging Lenses

• Converging (convex) lenses are thicker in the middle, converging parallel light rays to a focal point, and have a positive focal length.
• Diverging (concave) lenses are thinner in the middle, diverging parallel light rays from a focal point, and have a negative focal length.
• Converging lenses form both real and virtual images, depending on object distance.
• Diverging lenses always form virtual, upright, and reduced images.
• Lens Power (P) = 1/f (in meters), measured in diopters (D); converging lenses have positive power, diverging lenses have negative power.