Light striking a smooth surface obeys two simple laws — yet from them fall out every image a plane, concave or convex mirror can make. We derive the mirror equation from scratch, then let you drag the object through every case on a live bench.
The angle of incidence equals the angle of reflection, both measured from the normal at the point of incidence: ∠i = ∠r.
The incident ray, the reflected ray and the normal all lie in one plane. On a curved mirror the normal is simply the line drawn to the centre of curvature C.
For a plane mirror these two laws already fix the image: rays from a point source, on reflection, appear to diverge from a single point exactly as far behind the mirror as the source is in front. That image is virtual, erect and the same size — switch the bench below to plane to see the construction.
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Every formula on this page is written in one convention. Get these four rules right and the algebra takes care of real vs. virtual by itself.
Drag to orbit in 3D, or flip to the 2D ray diagram. Slide the object, switch mirrors, and hit a preset to jump between cases. Arrows nudge · P presenter · C clean-view · R reset (while hovering the bench).
Take a paraxial ray from an on-axis object O striking a concave mirror at A, and a second ray straight down the axis. Let α, β, γ be the angles AOP, ACP, AIP. The exterior-angle theorem plus the small-angle approximation does the rest.
Send the object to infinity: the incident rays are parallel and the image lands at the focus, so u → ∞ and v = f. The mirror equation gives 1/f = 2/R, i.e. the focus sits exactly midway between pole and centre of curvature.
The chief ray to the pole reflects with ∠BPA = ∠APB′, making triangles ABP and A′B′P similar. With signs, h₂/h₁ = −v/u.
Similar triangles give A′B′/AB = PA′/PA. Substituting A′B′ = h₂, AB = h₁, PA′ = −v, PA = −u yields m = h₂/h₁ = −v/u. A negative m means an inverted image; |m| > 1 means enlarged.
Tap any row to send the bench above to that case.
A convex mirror is simpler: for a real object it always makes a virtual, erect, diminished image between P and F. A plane mirror is the R → ∞ limit — virtual, erect, same size, equidistant behind.
The mirror equation holds more than static pictures. Differentiate it and it tells you how fast images move; square the magnification and it tells you how a solid object is stretched along the axis. These are the results JEE Advanced leans on.
Differentiate 1/v + 1/u = 1/f with f fixed: −dv/v² − du/u² = 0. Along the axis this gives
Motion across the axis scales only by m: v⊥,image = m · v⊥,object. Near the focus m is huge, so a slow object throws a blazing-fast image — the optical-lever trick.
Because dv/du = −m², a short rod lying along the axis images to a rod of length
Transverse scale m, axial scale m² — so a small sphere near a concave mirror images to a squashed egg, reversed front-to-back.
Concave: shaving & makeup mirrors (object inside F → magnified, erect), headlamps and torches (source at F → parallel beam), reflecting telescopes. Convex: vehicle and shop mirrors (wide, erect field of view).
The flaw: a real sphere shows spherical aberration — rim rays focus nearer the pole than paraxial rays. A parabolic mirror removes it, which is why telescope and torch mirrors are parabolas.
A concave mirror has focal length 15 cm. An object stands 25 cm in front of it. Find the image distance, magnification and nature.
Signs: u = −25 cm, f = −15 cm. From 1/v = 1/f − 1/u = −1/15 + 1/25 = −2/75, so v = −37.5 cm (in front → real).
m = −v/u = −(−37.5)/(−25) = −1.5.
The image is real, inverted and 1.5× enlarged, 37.5 cm in front of the mirror. (This is the C→F case — try it on the bench.)
A convex mirror of radius of curvature 30 cm has an object 10 cm in front. Locate the image and give its nature.
Convex: f = +R/2 = +15 cm, u = −10 cm. 1/v = 1/f − 1/u = 1/15 + 1/10 = 1/6, so v = +6 cm (behind → virtual).
m = −v/u = −6/(−10) = +0.6.
Image is virtual, erect and diminished, 6 cm behind the mirror — as always for a convex mirror.
A warm-up on the basics, then a JEE Advanced bank that pushes into image velocity, aberration and the tricky sign cases. Keep the sign convention in hand.