Formula sheet: Ray Optics And Optical Instruments

Ray Optics and Optical Instruments16 min readLesson 1 of 6

1.1 Mirror and lens formulas A spherical mirror links three signed lengths: the object and image positions measured from the pole and the focal length. The Cartesian convention decides which of them carry a minus sign, and the formula then holds without special cases. The magnification relates the image height to the object height. For a thin lens the same roles are played by the lens formula and the lens magnification. Signs decide most of the marks in this chapter. Distances measured along the direction of the incident light are positive, distances measured against it are negative, and heights measured upward from the principal axis are positive.

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}, m = \frac{h ^{'}}{h} = - \frac{v}{u}

Mirror formula and mirror magnification with signs

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}, m = \frac{v}{u}

Thin lens formula and lens magnification

P = \frac{1}{f}, \frac{1}{F} = \frac{1}{f _{1}} + \frac{1}{f _{2}}

Power in dioptres and the focal length of lenses in contact

1.2 Refraction, critical angle and the lens maker formula Snell's law connects the angles on the two sides of a refracting surface through the refractive indices. Light going from a denser to a rarer medium bends away from the normal, and beyond a critical angle it is totally internally reflected. The critical angle depends only on the two indices. The lens maker formula predicts the focal length of a thin lens from its refractive index and the radii of curvature of its faces. The same lens behaves differently in another medium, which is a standard numerical question.

n_{1} sin i = n_{2} sin r, n = \frac{c}{v}

Snell's law and the definition of refractive index

sin C = \frac{1}{n}, n_{1} sin C = n_{2}

Critical angle for total internal reflection

\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}

Refraction at a spherical surface

\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})

Lens maker formula in air

1.3 Prism, dispersion and optical instruments Inside a prism the deviation of a ray changes with the angle at which the ray meets the first face and with the angle between the two faces. At minimum deviation the ray passes symmetrically, which gives the prism formula used to measure the refractive index. Optical instruments combine lenses and mirrors. A compound microscope uses a short focal length objective to form a magnified real image, and the eyepiece magnifies it further. A telescope needs a large objective focal length to give angular magnification.

A + δ = i + e, n = \frac{sin ( \frac{A + δ _{m}}{2} )}{sin \frac{A}{2}}

Prism formula and the minimum deviation result

M = \frac{f _{o}}{f _{e}} (1 + \frac{f _{e}}{D})

Telescope magnification for the final image at the near point

M \approx \frac{f _{o}}{f _{e}} (final image at infinity), L = f_{o} + f_{e}

Telescope magnification at infinity and tube length

Exam

Write the sign convention values first, then substitute. A mirror question with one wrong sign loses more marks than a slow calculation done correctly.