Ray optics (geometrical optics) treats light as rays that travel in straight lines in homogeneous media, obeying the laws of reflection and refraction. It successfully explains the working of mirrors, lenses, prisms and optical instruments.
Reflection at Curved Surfaces (Mirrors)
Mirror Formula: 1/v + 1/u = 1/f = 2/R, where u = object distance, v = image distance, f = focal length, R = radius of curvature.
Magnification (linear): m = -v/u. If m is negative, image is inverted.
Sign convention (New Cartesian): distances measured from the pole; distances in direction of incident light are positive.
Refraction
Snell's Law: n1 sin(theta1) = n2 sin(theta2). The refractive index n = c/v.
Absolute refractive index relates speed of light in medium to speed in vacuum.
Critical Angle and Total Internal Reflection (TIR): When light goes from denser to rarer medium, if angle of incidence > critical angle thetac, all light reflects internally. sin(thetac) = n2/n1. TIR is used in optical fibres and diamonds.
Refraction at a spherical surface: n2/v - n1/u = (n2 - n1)/R
Lenses
Lens Maker's Equation: 1/f = (n - 1)(1/R1 - 1/R2), where n is refractive index of lens material.
Thin Lens Formula: 1/v - 1/u = 1/f
Magnification: m = v/u
Power of lens: P = 1/f (in dioptres when f in metres)
Combination of thin lenses in contact: 1/f = 1/f1 + 1/f2; P = P1 + P2
Prism
Deviation by prism: delta = (i + e) - A, where A is prism angle, i = angle of incidence, e = angle of emergence.
Minimum deviation condition: i = e and r1 = r2 = A/2.
At minimum deviation: n = sin((A + Dm)/2) / sin(A/2)
Optical Instruments
Simple Microscope: m = 1 + D/f (for image at near point), where D = 25 cm.
Compound Microscope: M = me x mo = -(L/fo) x (1 + D/fe), where L is tube length.
Astronomical Telescope (at relaxed eye): M = -fo/fe (angular magnification).
Reflecting Telescope: Uses concave mirror instead of objective lens; no chromatic aberration.
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An object is placed 30 cm from a concave mirror of focal length 20 cm. Find image distance and magnification.
1/v = 1/f - 1/u = 1/(-20) - 1/(-30) = -1/20 + 1/30 = -3/60 + 2/60 = -1/60. v = -60 cm. m = -v/u = -(-60)/(-30) = -2. Image is real, inverted, magnified twice.
Find critical angle for glass-air interface if nglass = 1.5.
sin(thetac) = nair/nglass = 1/1.5 = 2/3. thetac = sin-1(2/3) = 41.8 degrees.
A convex lens of focal length 20 cm forms an image. Object is at 30 cm. Find image position.
1/v - 1/u = 1/f. u = -30 cm (using sign convention), f = 20 cm.
1/v = 1/20 + 1/(-30)... Wait: 1/v = 1/f + 1/u = 1/20 + (-1/30)... Actually 1/v - 1/u = 1/f => 1/v = 1/f + 1/u = 1/20 + 1/(-30) = 3/60 - 2/60 = 1/60. v = 60 cm. Image is real, on other side.
A glass prism (A = 60 degrees, n = 1.5) is at minimum deviation. Find Dm.
n = sin((A + Dm)/2) / sin(A/2). 1.5 = sin((60 + Dm)/2) / sin(30). sin(30) = 0.5. So sin((60 + Dm)/2) = 1.5 x 0.5 = 0.75. (60 + Dm)/2 = 48.6 degrees. Dm = 2 x 48.6 - 60 = 37.2 degrees.
A compound microscope has fo = 2 cm, fe = 5 cm, tube length L = 15 cm. Find magnification (image at 25 cm).
M = (L/fo) x (1 + D/fe) = (15/2) x (1 + 25/5) = 7.5 x 6 = 45.
Two lenses of power +3 D and -1 D are in contact. Find equivalent focal length.
P = P1 + P2 = 3 + (-1) = 2 D. f = 1/P = 0.5 m = 50 cm.
An astronomical telescope has objective of f = 100 cm and eyepiece f = 5 cm. Find magnifying power in normal adjustment.
M = -fo/fe = -100/5 = -20. Magnitude is 20; negative sign indicates inverted image.
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Common mistakes
- Forgetting the sign convention (New Cartesian) when substituting values.
- Confusing the formula for simple microscope (m = D/f or 1 + D/f) depending on image position.
- Using degree instead of radian for angular calculations in telescopes.
Summary
Ray optics uses straight-line propagation, reflection and refraction laws to analyse mirrors, lenses and prisms. Key formulae: mirror (1/v + 1/u = 1/f), lens (1/v - 1/u = 1/f), prism (minimum deviation condition). Optical instruments exploit these to magnify images.