Optics

Essential formulas and concepts in optics

Snell’s Law (Refraction)

\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)

Variables:

Description/Usage:

Mathematical concept visualization

Thin Lens/Mirror Equation

\(\displaystyle \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)

Variables:

  • \(f = focal length of a lens or mirror, d_o = object distance (distance from object to lens/mirror), d_i = image distance (distance from lens/mirror to the image formed). (Sign conventions apply for virtual images, etc.)\)

Description/Usage:

Relates object distance, image distance, and focal length for paraxial rays. It works for converging lenses/mirrors (positive \(f\)) and diverging (using sign conventions with negative \(f\)). Often used with magnification formula for locating images.

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Magnification

\(m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\)

Variables:

  • \(h_o = object height, h_i = image height, d_o = object distance, d_i = image distance. The negative sign indicates that a positive m (image upright) comes with a negative d_i in sign convention (virtual image for lenses, etc.), and a negative m means the image is inverted.\)

Description/Usage:

The magnification \(m\) tells how large the image is relative to the object and whether it’s inverted (negative \(m\)) or upright (positive \(m\)). For example, \(|m|=2\) means the image is twice as tall as the object.

Mathematical concept visualization

Lensmaker’s Formula

\(\displaystyle \frac{1}{f} = (n-1)\Big(\frac{1}{R_1} - \frac{1}{R_2}\Big)\)

Variables:

  • \(f = focal length of a thin lens in air, n = refractive index of lens material, R_1 and R_2 = radii of curvature of the two lens surfaces (convention: R_1 is radius of surface facing the object, positive if center of curvature is on opposite side of incoming light).\)

Description/Usage:

Determines \(f\) from lens shape and material. For example, a symmetric bi-convex lens (\(R_1 = R_2 = R\)) in air has \(1/f = 2(n-1)/R\). It explains how stronger curvature (smaller \(R\)) or higher \(n\) gives a shorter focal length (more powerful lens).

Mathematical concept visualization