Optics

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mirrorImage Apas Cloud

The image a spherical mirror forms: 1/v + 1/u = 1/f in the Cartesian convention (light left to right, distances from the pole, positive along the light: a real object u < 0, a concave mirror f < 0, a convex mirror f > 0). The record has `v`, `m` = -v/u, `nature` in words ("real, inverted", "virtual, erect", "at infinity"), `real` and `erect`. An object at the focus gives `v` and `m` as the word `infinity`.

u (number) -- the object distance (a real object negative)
f (number) -- the focal length (concave negative, convex positive)

lensImage Apas Cloud

The image a thin lens forms: 1/v - 1/u = 1/f, the same convention (a convex lens f > 0, a concave lens f < 0; a real image beyond the lens has v > 0). The record has `v`, `m` = v/u, `nature`, `real`, `erect`; an object at the focus gives `infinity`.

u (number) -- the object distance (a real object negative)
f (number) -- the focal length (convex positive, concave negative)

lensMaker Apas Cloud

The lens maker's formula 1/f = (n - 1)(1/R1 - 1/R2) for a thin lens of index n in air; R1 is the first surface the light meets (convex towards the light positive), R2 the second. A flat face is the word "flat". The focal length, or `infinity` for a plane slab.

n (number) -- the refractive index of the lens relative to its surroundings
R1 (number or "flat") -- the first surface's radius of curvature
R2 (number or "flat") -- the second surface's radius of curvature

snell Apas Cloud

Snell's law n1 sin i = n2 sin r: the angle of refraction in degrees for a ray at incidence i from medium n1 into n2, or the words `total internal reflection` when n1 sin i exceeds n2.

n1 (number) -- the index of the medium the ray comes from
n2 (number) -- the index of the medium it enters
i (number) -- the angle of incidence in degrees, 0 to 90

criticalAngle Apas Cloud

The critical angle in degrees for light in the denser medium n1 meeting the rarer n2: sin C = n2/n1. Words when n1 is not the denser medium (no total internal reflection from that side).

n1 (number) -- the index of the denser medium the light is in
n2 (number) -- the index of the medium beyond

prismDeviation Apas Cloud

The ray through a prism of angle A and index n entering at incidence i: `r1` the refraction at the first face, `r2` = A - r1 the incidence at the second, `e` the emergence and `delta` = i + e - A the deviation, all in degrees. When r2 exceeds the critical angle the ray is reflected inside: `totalInternalReflection` is true and `e` and `delta` are the words.

A (number) -- the prism's refracting angle in degrees
i (number) -- the angle of incidence at the first face in degrees
n (number) -- the index of the prism relative to its surroundings

minimumDeviation Apas Cloud

A prism's minimum deviation, when the ray inside runs parallel to the base (r1 = r2 = A/2): `delta` = 2 asin(n sin(A/2)) - A and the incidence `i` that gives it, in degrees.

A (number) -- the prism's refracting angle in degrees
n (number) -- the index of the prism relative to its surroundings

prismIndex Apas Cloud

The refractive index from a prism's angle and its minimum deviation: n = sin((A + delta)/2) / sin(A/2), the measurement a spectrometer makes.

A (number) -- the prism's refracting angle in degrees
deltaMin (number) -- the minimum deviation in degrees

sphericalRefraction Apas Cloud

Refraction at one spherical surface from medium n1 into n2: n2/v - n1/u = (n2 - n1)/R, R positive when the centre of curvature lies on the outgoing side. A flat surface is the word "flat" (then v = n2 u / n1, the apparent depth). The image distance v, or `infinity`.

n1 (number) -- the index the light comes from
n2 (number) -- the index it enters
u (number) -- the object distance (a real object negative)
R (number or "flat") -- the surface's radius of curvature

imageChain Apas Cloud

The image through thin lenses and mirrors along one axis, in the order the light meets them (light starts rightwards). Each element is a map `{"kind": "lens" or "mirror", "x": its position, "f": its focal length}` (f signed for light arriving from the left). The object at `x` with height `h` (1 unless given). A list of stages, one record an element: `u` and `v` from that element along the light, `m`, the image's `x` and `h`, `mTotal`, `direction` after it (+1 right, -1 left after a mirror), `real`, `atInfinity` (then the chain stops there).

elements (list of maps) -- the lenses and mirrors as {kind, x, f}
x (number) -- the object's position on the axis
h (number) -- the object's height (optional, 1)

microscope Apas Cloud

The compound microscope: objective fo and eyepiece fe (both positive), the object at uo (negative, beyond fo so the objective forms a real magnified image). The final image at infinity (the default, `"infinity"`) or at the near point D (`"near-point"`, D 25 unless given). The record: `m` (mo times me, negative for inverted), `length` (objective to eyepiece), `vo`, `mo`, `ue`, `ve`, `me` (D/fe or 1 + D/fe) and `adjustment`. NCERT Example 9.9: fo 1.25, fe 5, uo -1.5 at the near point gives m = -30 and a tube of 11.67.

fo (number) -- the objective's focal length
fe (number) -- the eyepiece's focal length
uo (number) -- the object distance from the objective (negative)
adjustment (text) -- "infinity" (the default) or "near-point" (optional)
D (number) -- the near-point distance (optional, 25)

telescope Apas Cloud

The astronomical telescope on a distant object: objective fo, eyepiece fe. In normal adjustment (`"infinity"`, the default) m = -fo/fe and the length is fo + fe; with the final image at the near point D (`"near-point"`, D 25 unless given) m = -(fo/fe)(1 + fe/D) and the length fo + fe D/(fe + D). The record: `m`, `length`, `vo` = fo, `ue`, `ve`, `me`, `adjustment`. NCERT Example 9.10: fo 144, fe 6 gives m = -24 and a length of 150.

fo (number) -- the objective's focal length
fe (number) -- the eyepiece's focal length
adjustment (text) -- "infinity" (the default) or "near-point" (optional)
D (number) -- the near-point distance (optional, 25)

mirrorOn Apas Cloud

A spherical mirror's ray diagram from one call, the image where `mirrorImage(u, f)` puts it: the principal axis, the mirror as an arc of radius 2f about C with its back hatched (concave f < 0, convex f > 0), P, F and C marked, the object arrow at u (a real object, u < 0) of height h (0.3 |f| unless given), the three principal rays from its tip (parallel to the axis, through or towards C, through or towards F) with their arrowheads mid-ray, meeting at a real image or traced back dashed to a virtual one, the image arrow (dashed when virtual), and a line with u, f, v, m and the nature. When the diagram has no ranges yet they are set round the picture and the axes and grid turned off. The diagram's summary is `v`, `m` and the nature, so a `{{d}}` in prose shows it. An object at F gives parallel reflected rays and the note `image at infinity`.

d (diagram) -- the diagram to draw on
f (number) -- the focal length (concave negative, convex positive)
u (number) -- the object distance, negative
h (number) -- the object's height (optional)

lensOn Apas Cloud

A thin lens's ray diagram, the image where `lensImage(u, f)` puts it: the axis, the lens outline (convex f > 0, concave f < 0), O, F, F', 2F and 2F' marked, the object arrow at u (real, u < 0), the ray through O, the ray parallel to the axis refracted through F' (or as if from F for a concave lens) and the ray through F leaving parallel, the image arrow (dashed when virtual, with the rays traced back to it), and a line with u, f, v, m and the nature; ranges, axes and the summary as `mirrorOn`.

d (diagram) -- the diagram to draw on
f (number) -- the focal length (convex positive, concave negative)
u (number) -- the object distance, negative
h (number) -- the object's height (optional)

refractionOn Apas Cloud

Refraction at a plane surface: the interface across the picture (medium n1 above, n2 below, the denser shaded), the dashed normal, the incident ray at i from the upper left, the refracted ray at r from `snell(n1, n2, i)` with the angle arcs i and r, the weak partial reflection dashed; above the critical angle the reflected ray is drawn strong, the refracted one not at all, and the line says `total internal reflection`. The summary is r, or the words.

d (diagram) -- the diagram to draw on
n1 (number) -- the index above the surface, where the ray comes from
n2 (number) -- the index below
i (number) -- the angle of incidence in degrees

slabOn Apas Cloud

A ray through a rectangular slab of index n and thickness t in air: in at i, inside at r = `snell(1, n, i)`, out at e = i parallel to the way it came, the normals at both faces dashed, the undeviated path dashed and the lateral shift t sin(i - r)/cos r marked between the two with its value. The summary is r and the shift.

d (diagram) -- the diagram to draw on
n (number) -- the slab's index
t (number) -- the slab's thickness
i (number) -- the angle of incidence in degrees

prismRaysOn Apas Cloud

A prism's ray diagram: the triangular section with its apex up and the angle A written, the ray in at the left face at incidence i to the dashed normal, inside at r1, meeting the second face at r2, out at e (the angles from `prismDeviation(A, i, n)`), the incident direction continued and the emergent ray traced back dashed to where they meet, the deviation δ marked there, and a line with every angle (and `minimum deviation` when i = e). When r2 exceeds the critical angle the ray is reflected inside to the base and the line says so. The summary is δ.

d (diagram) -- the diagram to draw on
A (number) -- the prism's refracting angle in degrees
i (number) -- the angle of incidence at the first face in degrees
n (number) -- the prism's index relative to its surroundings

microscopeOn Apas Cloud

The compound microscope's ray diagram as `microscope(fo, fe, uo[, adjustment[, D]])` lays it out: the objective at the origin and the eyepiece a tube length away, Fo and Fe marked, the object arrow at uo, two rays from its tip (through the objective's centre and parallel to the axis) meeting at the real inverted intermediate image I', then through the eyepiece: leaving parallel when the final image is at infinity (the default), or diverging as if from the large virtual final image I at the near point D (drawn dashed) with `"near-point"`. A line gives vo, L, the final image and m; the summary is m and L.

d (diagram) -- the diagram to draw on
fo (number) -- the objective's focal length
fe (number) -- the eyepiece's focal length
uo (number) -- the object distance from the objective, negative, beyond fo
adjustment (text) -- "infinity" (the default) or "near-point" (optional)
D (number) -- the near-point distance (optional, 25)

telescopeOn Apas Cloud

The astronomical telescope's ray diagram as `telescope(fo, fe[, adjustment[, D]])` lays it out: the objective at the origin, the eyepiece at the length, three parallel rays from a distant point at a small angle α meeting at the real inverted image I' in the objective's focal plane, then through the eyepiece leaving parallel at the larger angle β (normal adjustment, the default) or diverging from the virtual final image at the near point D with `"near-point"`. The lenses are drawn at a schematic size, as the books draw them. A line gives L and m; the summary is m and L.

d (diagram) -- the diagram to draw on
fo (number) -- the objective's focal length
fe (number) -- the eyepiece's focal length
adjustment (text) -- "infinity" (the default) or "near-point" (optional)
D (number) -- the near-point distance (optional, 25)

layerTrace Apas Cloud

A ray through parallel layers (JEE: stacked slabs, apparent depth): from a medium n0 (1 unless given) at incidence i into layers of the given indices and thicknesses, top first, leaving into n0 below. The record: `angles` (to the normal, one a layer), `exitAngle` (i again, or the words when the ray is totally reflected), `totalInternalReflectionAt` (the layer number, or "none"), `shiftAlong` (how far the ray moves sideways through the stack), `lateralShift` (perpendicular to the emergent ray), `apparentDepth` (the sum of t/(n/n0), a point at the bottom seen from above) and `realDepth`.

indices (list) -- the layers' refractive indices, top first
thicknesses (list) -- the layers' thicknesses
i (number) -- the angle of incidence in degrees
n0 (number) -- the index of the medium above and below (optional, 1)

tirRadius Apas Cloud

A point source at the given depth in a medium of index n under air: the radius of the circle on the surface through which its light escapes, depth tan C = depth / sqrt(n^2 - 1) (the fish's circle of view).

n (number) -- the medium's index, above 1
depth (number) -- the source's depth below the surface

silveredLens Apas Cloud

A thin lens of index n with faces R1 (first met) and R2, one face silvered ("first" or "second"; the light comes from the unsilvered side): the light passes the lens, reflects and passes back, so the whole acts as a mirror of power P = 2/fLens - 1/fMirror with F = -1/P (a converging combination has F < 0, as a concave mirror). The silvered face's mirror focal length is R/2 with the lens maker's signs; a plano-convex lens silvered on its curved face gives |F| = R/2n, on its flat face |F| = fLens/2. The record: `F`, `fLens`, `fMirror`, and the image of an object at u: `v`, `m`, `nature`. A flat face is "flat".

n (number) -- the lens's index
R1 (number or "flat") -- the first face's radius
R2 (number or "flat") -- the second face's radius
side (text) -- "first" or "second": the silvered face
u (number) -- the object distance (negative)

compositeLens Apas Cloud

Thin lenses cemented in contact (JEE: two plano-convex halves of different glass, a biconvex lens cut along its axis): each part a map {"n", "R1", "R2"} (a face "flat" allowed), its focal length by the lens maker's formula and the combination 1/f = sum 1/f_k. The record: `f` and the list `parts`.

parts (list of maps) -- the parts as {n, R1, R2}, in the order the light meets them

mirrorImages Apas Cloud

The number of images of a point object between two plane mirrors at the given angle: with n = 360/theta whole, n - 1 when n is even, or when the object is on the bisector ("bisector" as the second argument); n when n is odd and the object is off it; otherwise the whole part of n.

theta (number) -- the angle between the mirrors in degrees
position (text) -- "bisector" when the object lies on the bisector (optional)

fibreTrace Apas Cloud

An optical fibre (a core of index n1 in a cladding n2): a ray entering the end face from air at i refracts to `theta` from the axis and meets the wall at `atWall` from its normal; `guided` when that exceeds the `critical` angle asin(n2/n1); the `acceptance` angle (sin i_max = sqrt(n1^2 - n2^2)), the `numericalAperture`, and with a length L and a core width the number of `reflections`.

n1 (number) -- the core's index
n2 (number) -- the cladding's index, below n1
i (number) -- the entry angle from air in degrees
L (number) -- the fibre's length (optional, with width)
width (number) -- the core's width (optional, with L)

layersOn Apas Cloud

The JEE stacked-slab figure from one call: the layers shaded with their index and thickness written in, the ray in from above at i (the medium above n0, 1 unless given), through each layer at its own angle from `layerTrace` with dashed normals and the angles written, out below parallel to the way in with the undeviated path dashed; where the sine passes 1 the ray is totally reflected at that layer's top and goes back up. The caption gives the lateral shift and the apparent depth; the summary too.

d (diagram) -- the diagram to draw on
indices (list) -- the layers' indices, top first
thicknesses (list) -- the layers' thicknesses
i (number) -- the angle of incidence in degrees
n0 (number) -- the index above and below (optional, 1)

tirCircleOn Apas Cloud

A point source at the given depth in a medium of index n under air: the medium shaded, the source S, the critical rays to the edge of the escape circle grazing out along the surface, a ray inside the cone refracting out, one beyond C reflected back, the circle's radius R = depth tan C marked. The summary is C and R.

d (diagram) -- the diagram to draw on
n (number) -- the medium's index, above 1
depth (number) -- the source's depth

sphericalSurfaceOn Apas Cloud

Refraction at one spherical surface from n1 into n2 (JEE: a glass rod's curved end, a sphere's face): the surface as an arc of radius R through the pole P (centre C on the outgoing side when R > 0; "flat" for a plane face), medium 2 shaded, the object arrow at u (negative) of height h, the ray through C undeviated and the ray parallel to the axis refracted at the surface, the image where `sphericalRefraction(n1, n2, u, R)` puts it (dashed when virtual, m = n1 v / (n2 u)). The summary is v and m.

d (diagram) -- the diagram to draw on
n1 (number) -- the index the light comes from
n2 (number) -- the index beyond the surface
R (number or "flat") -- the surface's radius of curvature
u (number) -- the object distance, negative
h (number) -- the object's height (optional)

lensSystemOn Apas Cloud

Lenses and mirrors in combination on one axis (JEE: two lenses a distance apart, a lens with a mirror behind it): the elements as `imageChain` takes them, each drawn at its x (a lens outline, convex or concave by the sign of f; a mirror arc facing the light), their separations marked under the axis, the object at u (its position on the axis, the first element usually at 0), one ray a stage through the element's centre (or to a mirror's pole) on to that stage's image, each intermediate image a lighter arrow I1, I2, ..., the final image A'B' (dashed when virtual); an image at infinity ends the chain with parallel rays. The caption and summary give the final image's position and the whole magnification.

d (diagram) -- the diagram to draw on
elements (list of maps) -- the lenses and mirrors as {kind, x, f}
u (number) -- the object's position on the axis, before the first element
h (number) -- the object's height (optional)

silveredLensOn Apas Cloud

A lens with one face silvered, as `silveredLens` works it out: the lens outline from its two radii with the silvered face thick and hatched on the outside, the equivalent mirror's focus F marked, the object at u, the ray to the centre and the ray parallel to the axis coming back to the image (dashed when virtual). The caption gives f_l, f_m and F, then v and m; the summary F, v and m.

d (diagram) -- the diagram to draw on
n (number) -- the lens's index
R1 (number or "flat") -- the first face's radius
R2 (number or "flat") -- the second face's radius
side (text) -- "first" or "second": the silvered face
u (number) -- the object distance, negative
h (number) -- the object's height (optional)

compositeLensOn Apas Cloud

A lens of two or more materials cemented (JEE: cut halves, a plano-convex on a plano-concave): each part's outline from its radii side by side with the interfaces dashed and the indices written above, the combined focal length from `compositeLens` in the caption and the summary; with an object at u the image by `lensImage(u, f)` and two rays.

d (diagram) -- the diagram to draw on
parts (list of maps) -- the parts as {n, R1, R2}
u (number) -- the object distance, negative (optional)
h (number) -- the object's height (optional)

planeMirrorsOn Apas Cloud

Two plane mirrors from a vertex at the given angle, their backs hatched, a point object O on the dashed circle at the given angle from the first mirror (a third of the angle unless given), every image by reflection in the mirrors (at 2k theta ± the object's angle, outside the wedge) as I1, I2, ..., the first two with their dashed construction lines; the count from `mirrorImages` in the caption and the summary.

d (diagram) -- the diagram to draw on
theta (number) -- the angle between the mirrors in degrees
objectAngle (number) -- the object's angle from the first mirror (optional)

fibreOn Apas Cloud

An optical fibre of the given length and core width: the core between two cladding strips, the dashed axis, the ray in from air at i refracting to its angle inside, reflected at each wall when guided (the count from `fibreTrace` in the caption; when more than ten legs would fit, the first legs are drawn to scale and the true count written), or refracted out into the cladding at the first wall when the angle there is below the critical angle. The summary says guided with the count, or not guided.

d (diagram) -- the diagram to draw on
n1 (number) -- the core's index
n2 (number) -- the cladding's index
i (number) -- the entry angle from air in degrees
L (number) -- the fibre's length
width (number) -- the core's width