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Beyond Advanced · Ray Optics

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1
A plane mirror is the segment x=0x=0, 0≤y≤40\le y\le4 (metres). A point source is at (3,0)(3,0) and an eye moves along (2,Y)(2,Y). A black screen at x=1x=1 covers 2≤y≤32\le y\le3. Only paths source-mirror-eye count; touching an opaque edge is blocked. Find the total length, in metres, of the allowed set of eye coordinates YY. Give the numerical answer to two decimal places.
Not attempted
source (3,0)eye line x = 2mirror x = 0screen
Your answer: —
Correct answer: 3.75
2
Three infinite plane mirrors are x=0x=0, y=0y=0, and x+y=2ax+y=2a, where a>0a>0. An image is constructed by reflecting a point successively in that stated order. Which map sends an initial point (x,y)(x,y) to the final image?
Not attempted
A
(2a+y,2a+x)(2a+y,2a+x)
B
(2a−x,2a−y)(2a-x,2a-y)
C
(y,x)(y,x)
D
(2a−y,2a−x)(2a-y,2a-x)
3
An infinite plane mirror rotates about the origin in the xyxy plane with constant angular speed 2 rad s−12\text{ rad s}^{-1}. At t=0t=0 it is the xx axis. A point object is then at (3,4) m(3,4)\text{ m}, moving with constant laboratory velocity (1,2) m s−1(1,2)\text{ m s}^{-1}. Find the magnitude of the instantaneous image acceleration in m s−2\text{m s}^{-2}, to two decimal places. Ignore propagation delay.
Not attempted
Your answer: —
Correct answer: 78.79
4
Two plane mirrors through the origin have line angles α(t)\alpha(t) and β(t)\beta(t) relative to the xx axis. A collimated incident beam has direction angle ϕ(t)\phi(t). It reflects first at the first mirror and then at the second; the geometry allows both bounces. All angular quantities are signed and continuously unwrapped. Select all correct statements.
Not attempted
A
If α˙=β˙\dot\alpha=\dot\beta, the output angular speed equals ϕ˙\dot\phi.
B
If ϕ˙=0\dot\phi=0, α˙=Ω\dot\alpha=\Omega and β˙=−Ω\dot\beta=-\Omega, the output angular speed is −4Ω-4\Omega.
C
Interchanging reflection order leaves the unwrapped output angle unchanged for every mirror angle.
D
For fixed incident direction, rotating both mirrors rigidly together changes the output direction at twice their common angular speed.
5
A point O=(a,b,c)O=(a,b,c), with all coordinates positive, is imaged by three mutually perpendicular plane mirrors x=0,y=0,z=0x=0,y=0,z=0. Consider the three single-reflection images, three double-reflection images and one triple-reflection image. What is the centre of mass of these seven equal-weight image points?
Not attempted
A
(a/7,b/7,c/7)(a/7,b/7,c/7)
B
(−a/7,−b/7,−c/7)(-a/7,-b/7,-c/7)
C
(0,0,0)(0,0,0)
D
(−a,−b,−c)(-a,-b,-c)
6
An eye is at (2,0)(2,0) and a luminous vertical segment at x=6x=6 spans −3≤y≤5-3\le y\le5, all lengths in metres. A plane mirror on x=0x=0 can be translated vertically. The eye must see the entire segment for every eye height −1≤yE≤2-1\le y_E\le2 while xE=2x_E=2 remains fixed. Find the minimum mirror length in metres. Give the numerical answer to two decimal places.
Not attempted
mirror x = 0segment x = 6eye x = 2yE from −1 to 2
Your answer: —
Correct answer: 4.25
7
An infinite plane mirror is the line x=s(t)x=s(t), and an object has coordinates (xO(t),yO(t))(x_O(t),y_O(t)). At a specified instant s=3s=3, s˙=2\dot s=2, s¨=−1\ddot s=-1, xO=1x_O=1, x˙O=−4\dot x_O=-4, x¨O=3\ddot x_O=3, y˙O=5\dot y_O=5, y¨O=2\ddot y_O=2, in SI units. Ignore propagation delay. Select all correct statements.
Not attempted
A
The image moves in the positive xx direction at 9 m s−19\text{ m s}^{-1}.
B
The separation of image and object is decreasing at 12 m s−112\text{ m s}^{-1}.
C
The image acceleration magnitude is 29 m s−2\sqrt{29}\text{ m s}^{-2}.
D
The image tangential velocity is −5 m s−1-5\text{ m s}^{-1}.
8
In three dimensions a ray with direction proportional to (1,2,−2)(1,2,-2) reflects first from a plane with normal proportional to (1,1,0)(1,1,0), then from a plane with normal proportional to (0,1,1)(0,1,1). The planes are positioned to allow that order. Which direction is proportional to the output direction?
Not attempted
A
(2,−2,1)(2,-2,1)
B
(−2,1,2)(-2,1,2)
C
(−2,2,1)(-2,2,1)
D
(−1,−2,2)(-1,-2,2)
9
A source and receiver are fixed at (0,3)(0,3) and (8,1)(8,1) metres. A horizontal mirror occupies y=0y=0, 5≤x≤75\le x\le7. An opaque vertical shutter at x=7x=7 transmits only through a small slit. Find the height in metres at which the slit must be centred for the single-reflection path to reach the receiver. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.5
10
Two mirrors meet at an angle θ\theta. An object lies strictly inside the wedge. Match each configuration in List I to the number of distinct geometrical images in List II; the usual ideal inclined-mirror construction applies. List IList IIP: 60 degrees, off bisector1: 4Q: 72 degrees, on bisector2: 5R: 72 degrees, off bisector3: 3S: 90 degrees, off bisector4: 65: 2\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: 60 degrees, off bisector}&\text{1: }4\\\text{Q: 72 degrees, on bisector}&\text{2: }5\\\text{R: 72 degrees, off bisector}&\text{3: }3\\\text{S: 90 degrees, off bisector}&\text{4: }6\\\text{}&\text{5: }2\end{array}
Not attempted
A
P-1, Q-2, R-3, S-2
B
P-2, Q-3, R-2, S-1
C
P-3, Q-2, R-1, S-2
D
P-2, Q-1, R-2, S-3
11
An object and eye are at (a,yO)(a,y_O) and (b,yE)(b,y_E) with a,b>0a,b>0, on one side of the mirror x=0x=0. Its finite reflective strip is L≤y≤UL\le y\le U. Treat the eye as a point. Select all correct statements.
Not attempted
A
For fixed eye, the minimum mirror length to see an object of height HH is bH/(a+b)bH/(a+b).
B
Doubling aa and bb leaves the field of view on the object line unchanged.
C
Translating the eye upward by Δ\Delta always translates the required strip upward by Δ/2\Delta/2.
D
If a=3ba=3b, the required strip is one fourth of the object height.
12
A plane mirror rotates anticlockwise at constant angular speed ω>0\omega>0 about the origin and is the xx axis at t=0t=0. A point object is then at (a,0)(a,0), a>0a>0, and moves with constant velocity (0,v)(0,v). Its image is instantaneously stationary. Which is the image jerk d3I/dt3d^3I/dt^3 at that instant? Ignore light-travel delay.
Not attempted
A
(0,16aω3)(0,16a\omega^3)
B
(0,0)(0,0)
C
(0,−8aω3)(0,-8a\omega^3)
D
(0,8aω3)(0,8a\omega^3)
13
A concave mirror of focal-length magnitude 12 cm12\text{ cm} faces left. Its pole is instantaneously at x=0x=0 and moves right at 2 cm s−12\text{ cm s}^{-1}. A point object is at x=−18 cmx=-18\text{ cm}, moving left at 1 cm s−11\text{ cm s}^{-1}. What is the laboratory image velocity along positive xx, in cm s−1\text{cm s}^{-1}? Ignore propagation delay. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectmirror f = −12 cm18 cm
Your answer: —
Correct answer: 14
14
A stationary concave mirror has focal-length magnitude f>0f>0. A straight luminous rod lies on its axis, its ends at distances pp and p+ℓp+\ell in front, with p>fp>f and ℓ>0\ell>0. For which pp is the real image rod exactly as long as the object rod?
Not attempted
A
p=2f−ℓ/2p=2f-\ell/2
B
p=f+ℓ2+4f2−ℓ2p=f+\dfrac{\sqrt{\ell^2+4f^2}-\ell}{2}
C
p=2fp=2f
D
p=f+f2+ℓ2p=f+\sqrt{f^2+\ell^2}
15
A stationary concave mirror of focal-length magnitude ff faces an axial luminous segment occupying every object distance ss from f/2f/2 to 3f/23f/2. Consider ideal paraxial geometrical images of all points except s=fs=f. Select all correct statements.
Not attempted
A
The finite images form two disconnected unbounded sets.
B
The shortest interval containing all finite images has length 4f4f.
C
No image point lies between the pole and v=f/2v=f/2.
D
Every point of the rod has a finite real image.
16
A concave mirror faces left with pole at x=0x=0 and focal-length magnitude 10 cm10\text{ cm}. A particle at (x,y)=(−30,2) cm(x,y)=(-30,2)\text{ cm} has instantaneous velocity (3,1) cm s−1(3,1)\text{ cm s}^{-1}. Find the magnitude of the image velocity in cm s−1\text{cm s}^{-1} in the paraxial approximation. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectmirror f = −10 cm30 cm
Your answer: —
Correct answer: 0.99
17
A concave mirror of focal-length magnitude ff is at x=0x=0, facing left. An opaque screen at x=−3fx=-3f has a very small hole on the axis. A small off-axis object at (x,y)=(−p,h)(x,y)=(-p,h), with p>3fp>3f and h≠0h\ne0, emits rays toward the mirror. In the paraxial limit, which pp makes the returning image lie in the screen plane?
Not attempted
A
p=9f/2p=9f/2
B
p=4fp=4f
C
No allowed value of pp exists.
D
p=6fp=6f
18
A concave spherical mirror and an equiconvex thin lens both have focal-length magnitude ff in air. They are separately immersed in a homogeneous liquid of index 4/34/3. The lens glass index is 3/23/2 and does not change. Each receives light from a real object at distance 3f3f in its own surrounding medium. Select all correct statements.
Not attempted
A
The mirror image is real with magnification −1/2-1/2.
B
The lens image is virtual with magnification +4+4.
C
The lens image is real at 12f12f.
D
The mirror focal-length magnitude becomes 4f4f.
19
A convex mirror has focal length +15 cm+15\text{ cm} with pole at x=0x=0. Two particles are instantaneously at x=−15 cmx=-15\text{ cm} and x=−45 cmx=-45\text{ cm}, moving right with speeds 44 and 8 cm s−18\text{ cm s}^{-1} respectively. Find the instantaneous rate of increase of the positive separation between their images, in cm s−1\text{cm s}^{-1}. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
particle: −45 cmparticle: −15 cmpole: 0; f = +15 cm
Your answer: —
Correct answer: 0.5
20
A real object forms a real image in a concave mirror. Its axial distance from the mirror is twice the image distance. The mirror is then translated directly towards the stationary object by one sixth of the original object distance. How does the transverse magnification change?
Not attempted
A
From −1/2-1/2 to −3/4-3/4.
B
From −1/2-1/2 to −1/3-1/3.
C
From −1/2-1/2 to +2/3+2/3.
D
From −1/2-1/2 to −2/3-2/3.
21
A concave mirror has focal-length magnitude ff. Match each specified object in List I to its signed transverse magnification in List II. Distances are positive physical distances in front of the mirror. List IList IIP: distance f/31: −2Q: distance 3f/22: −1/4R: distance 3f3: 3/2S: distance 5f4: −1/25: 2\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: distance f/3}&\text{1: }-2\\\text{Q: distance 3f/2}&\text{2: }-1/4\\\text{R: distance 3f}&\text{3: }3/2\\\text{S: distance 5f}&\text{4: }-1/2\\\text{}&\text{5: }2\end{array}
Not attempted
A
P-3, Q-1, R-4, S-2
B
P-1, Q-4, R-2, S-3
C
P-4, Q-2, R-3, S-1
D
P-2, Q-3, R-1, S-4
22
A concave mirror of focal-length magnitude 8 cm8\text{ cm} faces an axial rod. Its nearer end is 12 cm12\text{ cm} from the pole and its farther end is p>12 cmp>12\text{ cm} away. The real image rod has length 12 cm12\text{ cm}. Find pp, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
far end: pnear end: 12 cm|f| = 8 cmluminous axial rod
Your answer: —
Correct answer: 24
23
A spherical mirror is convex, has Cartesian f>0f>0, and receives a converging paraxial beam which would meet at a point a distance u>0u>0 behind the pole if the mirror were absent. Consider the reflected image with the standard mirror sign convention. Select all correct statements.
Not attempted
A
For u=f/2u=f/2, the reflected image is real and m=+2m=+2.
B
For u=2fu=2f, the reflected image is virtual and m=−1m=-1.
C
For u=fu=f, the reflected image is at infinity.
D
For every positive uu, the reflected image is diminished.
24
A paraxial axial particle moves in front of a stationary concave mirror of focal-length magnitude ff so that s(t)=f+k/(t+t0)s(t)=f+k/(t+t_0), where k,t0>0k,t_0>0 and t≥0t\ge0. Which description of the real image motion is correct? Let qq denote positive image distance in front of the mirror.
Not attempted
A
The image is virtual for all tt.
B
qq increases uniformly at speed f2/kf^2/k.
C
qq decreases uniformly at speed k/f2k/f^2.
D
qq has constant positive acceleration 2f2/k2f^2/k.
25
A thin converging lens of focal length 12 cm12\text{ cm} forms a real image of an object 18 cm18\text{ cm} in front. Between lens and image, normal parallel slabs of thicknesses 6 cm6\text{ cm} and 8 cm8\text{ cm}, indices 3/23/2 and 4/34/3, are inserted. They remain entirely before the final image and do not overlap. Find the final distance of the image from the lens, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlens f = 12 cmslab 1slab 218 cm
Your answer: —
Correct answer: 40
26
A point source is beneath two horizontal layers of thicknesses a,ba,b and indices n1,n2>1n_1,n_2>1, with air above. Let p=sin⁡ip=\sin i for the emergent ray in air. Its horizontal surface displacement from the source vertical is XX. Which exact relation follows from Snell refraction, without a paraxial approximation?
Not attempted
airlayer: b, n₂layer: a, n₁source
A
X=p(a+b)n1n2−p2X=\dfrac{p(a+b)}{\sqrt{n_1n_2-p^2}}
B
X=p(an1+bn2)X=p(an_1+bn_2)
C
X=p(an12−p2+bn22−p2)X=p\left(\dfrac{a}{\sqrt{n_1^2-p^2}}+\dfrac{b}{\sqrt{n_2^2-p^2}}\right)
D
X=p(a/n1+b/n2)X=p(a/n_1+b/n_2)
27
A ray traverses several parallel homogeneous layers, all bounded by horizontal planes. It starts in air and finally returns to air. Intermediate indices are at least one; every interface is reached before any lateral edge. The incident angle is 0<i<90∘0<i<90^\circ. Select all correct statements.
Not attempted
A
Total internal reflection cannot occur at any interface in this stack.
B
The output is parallel to the input.
C
Reordering the thickness-index pairs leaves the final lateral displacement unchanged.
D
Reordering those pairs leaves the transit time unchanged.
28
Two parallel transparent layers each have thickness 12 cm12\text{ cm}. Their indices are 5/45/4 and 5/35/3. A ray in air has sin⁡i=1\sin i=1, understood as the grazing limit i→90∘i\to90^\circ from below. Find the limiting horizontal displacement between the points where it enters the first layer and leaves the second, in centimetres. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 25
29
A lens of focal length ff is translated between a fixed object and screen separated by DD. A normal slab of thickness tt and index n>1n>1 is always entirely between the lens and the screen, attached to the screen. Its front face is far enough back to allow both sharp-image lens positions. Let dd be the separation of those positions. Which formula determines ff? The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectmovable lensslabscreenobject–screen: D
A
D2−d24D\dfrac{D^2-d^2}{4D}
B
(D+t−t/n)2−d24(D+t−t/n)\dfrac{(D+t-t/n)^2-d^2}{4(D+t-t/n)}
C
D2−d24(D−t+t/n)\dfrac{D^2-d^2}{4(D-t+t/n)}
D
(D−t+t/n)2−d24(D−t+t/n)\dfrac{(D-t+t/n)^2-d^2}{4(D-t+t/n)}
30
An object and screen are 90 cm90\text{ cm} apart. A 15 cm15\text{ cm} slab of index 3/23/2 lies next to the screen. A converging lens has two sharp-image positions 15 cm15\text{ cm} apart, both before the slab. Find the focal length in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectmovable lensslab: 15 cmscreen90 cm
Your answer: —
Correct answer: 20.59
31
A bird is a height hh above a horizontal water surface, and a fish is a depth dd below it. Water index is n>1n>1. Both are near the same vertical. Apparent distances are defined using paraxial rays around that vertical; ignore finite eye size. Select all correct statements.
Not attempted
A
The fish-to-bird apparent separation divided by the bird-to-fish apparent separation is n2n^2.
B
This ratio becomes one when h=dh=d.
C
For a fixed interface, the time derivatives of the two apparent separations have the same factor nn.
D
The bird must divide hh by nn as well as dd.
32
A parallel-sided slab of index n>1n>1 and normal thickness tt is in air. A ray enters with incidence ii. Define dd as the perpendicular separation of the outgoing ray and the unrefracted continuation of the incoming ray. As i→90∘i\to90^\circ, which pair gives the limits of dd and transit time inside the slab? Vacuum speed is cc.
Not attempted
A
(t,n2tcn2−1)\left(t,\dfrac{n^2t}{c\sqrt{n^2-1}}\right)
B
(t,∞)\left(t,\infty\right)
C
(0,nt/c)\left(0,nt/c\right)
D
(nt,ntcn2−1)\left(nt,\dfrac{nt}{c\sqrt{n^2-1}}\right)
33
A parallel slab in air is 10 cm10\text{ cm} thick. A ray enters with sin⁡i=3/5\sin i=3/5, cos⁡i=4/5\cos i=4/5. Its displacement perpendicular to the incoming ray is 2 cm2\text{ cm}. Find the slab index. Assume n>1n>1. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.34
34
A 12 cm12\text{ cm} normal slab is inserted completely before a paraxial focus in air. Match the slab index in List I to the downstream focus shift in centimetres in List II. List IList IIP: n=3/21: 6Q: n=4/32: 4R: n=23: 2S: n=6/54: 35: 8\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: n=3/2}&\text{1: }6\\\text{Q: n=4/3}&\text{2: }4\\\text{R: n=2}&\text{3: }2\\\text{S: n=6/5}&\text{4: }3\\\text{}&\text{5: }8\end{array}
Not attempted
A
P-3, Q-2, R-4, S-1
B
P-2, Q-4, R-1, S-3
C
P-4, Q-1, R-3, S-2
D
P-1, Q-3, R-2, S-4
35
A lens forms an image on a fixed screen. A normal slab of index n>1n>1 and thickness tt is inserted entirely between lens and original image. The paraxial focus remains beyond the slab. A second identical slab is instead placed wholly before the lens between object and lens. The physical object and lens positions remain fixed. Select all correct statements. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlensimage-side slabscreeninitial conjugates
A
An image-side slab shifts the focus downstream by Δ\Delta.
B
An object-side slab necessarily gives the same focus displacement Δ\Delta.
C
An object-side slab can turn a real image into a virtual image if s−Δ<fs-\Delta<f.
D
Zero angular deviation of a slab guarantees no change of lens conjugacy.
36
Two plane-parallel layers, thicknesses a,ba,b, have indices n1,n2n_1,n_2 and lie between air regions. At a given nonzero incident angle they have total ray transit time TT and total horizontal travel XX inside the stack. If their order is reversed while entrance height and incident direction are kept fixed, what happens?
Not attempted
A
XX is unchanged, but TT changes.
B
Both change unless n1=n2n_1=n_2.
C
Both TT and XX are unchanged.
D
TT is unchanged, but XX changes sign.
37
A point object in air is 20 cm20\text{ cm} before a normal glass slab of thickness 12 cm12\text{ cm} and index 3/23/2. A plane mirror is immediately behind its far face. Rays pass through the slab, reflect once, and return into the original air. Find the distance in centimetres of the final paraxial virtual image from the object. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectslab frontplane mirrorglass: 12 cm
Your answer: —
Correct answer: 56
38
A lens of focal length 10 cm10\text{ cm} is fixed 30 cm30\text{ cm} from an object in air. A normal 12 cm12\text{ cm} slab of index 3/23/2 can be placed either completely before or completely after the lens; sufficient clearance exists in both cases. What is the ratio of the final lens-to-image distance for object-side placement to that for image-side placement? The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlens f = 10 cm30 cm
A
11
B
19/1519/15
C
76/6576/65
D
65/7665/76
39
A lamp lies 12 cm12\text{ cm} below the boundary between a liquid of index 5/35/3 and an upper layer of index 5/45/4, thickness 9 cm9\text{ cm}. Air is above the upper layer. A centred opaque disk floats at the top. Find the minimum radius in centimetres that blocks every ray which can emerge into air. Ignore edge diffraction and include grazing limiting rays in the disk. Give the numerical answer to two decimal places.
Not attempted
air / opaque disk9 cm, n = 5/412 cm, n = 5/3lamp
Your answer: —
Correct answer: 21
40
A ray inside a transparent cube of index nn in air has unit direction (a,b,c)(a,b,c) with all components nonzero. It hits faces away from edges. What is the necessary and sufficient condition for it to undergo TIR at every face it encounters forever in the ideal ray model?
Not attempted
A
max⁡(a2,b2,c2)<1−1/n2\max(a^2,b^2,c^2)<1-1/n^2
B
min⁡(a2,b2,c2)<1−1/n2\min(a^2,b^2,c^2)<1-1/n^2
C
a2+b2+c2<1−1/n2a^2+b^2+c^2<1-1/n^2
D
max⁡(a2,b2,c2)>1/n2\max(a^2,b^2,c^2)>1/n^2
41
A ray is already inside an ideal transparent cube in air. It has direction proportional to (1,1,1)(1,1,1), and hits all three families of faces away from edges. Consider only geometric refraction and reflection; a critical-incidence ray is not counted as TIR. Select all correct statements.
Not attempted
A
For n=4/3n=4/3, the ray is trapped at every later face in the ideal model.
B
For n=3/2n=\sqrt{3/2}, all encounters count as TIR under the stated convention.
C
For 1<n<3/21<n<\sqrt{3/2}, each face admits a refracted ray.
D
A reflection can increase a later incidence angle by changing a component magnitude.
42
A transparent rectangular block has width 6 cm6\text{ cm} and height 8 cm8\text{ cm}, index 5/35/3, with air outside. A ray starts at (0,2) cm(0,2)\text{ cm} just inside the left face and travels into the block with direction proportional to (3,4)(3,4). Count the horizontal-face TIR bounces before the ray first leaves the block. Ignore partial reflection and follow only the refracted ray when TIR fails. Give the numerical answer to two decimal places.
Not attempted
6 cm8 cmstart (0,2)direction (3,4)
Your answer: —
Correct answer: 0
43
A homogeneous square of index nn is surrounded by air. An interior ray lies in the square’s plane and hits vertical and horizontal edges at incidence angles ii and 90∘−i90^\circ-i. For some direction to be strictly totally reflected at both edge families, what condition on nn is required?
Not attempted
A
n>2n>2
B
n>2n>\sqrt2
C
n≥2n\ge\sqrt2
D
n>1n>1
44
A meridional ray enters the flat end of a straight transparent rod from a liquid of index 6/56/5. The rod core index is 3/23/2 and its cylindrical wall is in a medium of index 2\sqrt2. Find the supremum of the entrance quantity 100sin⁡2θ100\sin^2\theta for strict TIR at the wall, where θ\theta is measured from the rod axis. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 17.36
45
A ray enters a plane-faced prism from air and later strikes a second face. Let the internal angles from the two face normals satisfy r1+r2=Ar_1+r_2=A, with 0≤r1,r2≤90∘0\le r_1,r_2\le90^\circ. The prism index is n>1n>1 and C=sin⁡−1(1/n)C=\sin^{-1}(1/n). Consider transmission through these two faces only, excluding grazing incidence and emergence. Select all correct statements.
Not attempted
Aincident raysecond faceprismschematic; angles given in text
A
A non-grazing two-face transmitted path exists if and only if A<2CA<2C.
B
For A=2CA=2C, a finite-angle symmetric transmitted path exists.
C
For A>2CA>2C, no ray in the specified two-face geometry emerges at the second face.
D
Failure to emerge at the second face means the ray energy is absorbed there.
46
A ray enters normally through the first short face of a right-isosceles prism of index 3/23/2. Its hypotenuse is in contact with a liquid of index nhn_h, while the other short face is in air. The internal geometry turns the ray by reflection at the hypotenuse before normal exit through the other short face. Which condition guarantees this route by strict TIR?
Not attempted
normal entryliquid onhypotenuseother short face: air
A
nh>3/(22)n_h>3/(2\sqrt2)
B
nh<2n_h<\sqrt2
C
nh<3/(22)n_h<3/(2\sqrt2)
D
nh<3/2n_h<3/2
47
A ray is launched upward through horizontal layers of indices 22, 3/23/2, and 5/45/4, finally reaching air. In the initial medium its angle to the vertical has sine 3/53/5. Which numbered interface first prevents a refracted ray: 1 is 2→3/22\to3/2, 2 is 3/2→5/43/2\to5/4, 3 is 5/4→15/4\to1? Follow the first upward passage only. Give the numerical answer to two decimal places.
Not attempted
airn = 5/4n = 3/2n = 2launch
Your answer: —
Correct answer: 3
48
Four internal rays of different transparent blocks strike an air interface. Match List I, specifying index nn and incidence sine ss, to the behaviour in List II. Ignore partial reflection when a refracted ray exists. List IList IIP: n=4/3, s=3/51: TIRQ: n=5/3, s=3/52: grazing emergenceR: n=2, s=3/53: transmitted sine 3/4S: n=3/2, s=1/24: transmitted sine 4/55: normal emergence\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: n=4/3, s=3/5}&\text{1: TIR}\\\text{Q: n=5/3, s=3/5}&\text{2: grazing emergence}\\\text{R: n=2, s=3/5}&\text{3: transmitted sine }3/4\\\text{S: n=3/2, s=1/2}&\text{4: transmitted sine }4/5\\\text{}&\text{5: normal emergence}\end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-3, R-4, S-2
C
P-3, Q-4, R-2, S-1
D
P-4, Q-2, R-1, S-3
49
A stationary spherical transparent body of radius RR has index n>1n>1 and is in air. A ray entering from air follows a chord inside. Let the internal chord’s perpendicular distance from the centre be bb. The body is homogeneous and interfaces are ideal. Select all correct statements.
Not attempted
A
A non-grazing ray entering from air cannot undergo strict TIR at its first exit.
B
An internally launched ray with b>R/nb>R/n can remain trapped by TIR.
C
Refraction at entrance can create a chord with b>R/nb>R/n for an air ray.
D
For an entering air ray, increasing external incidence can make the exit exceed the critical angle.
50
A lamp inside a sphere of radius RR and index n>1n>1 is at distance a<Ra<R from its centre. Disregard repeated partially reflected branches and follow each emitted ray to its first boundary. For every emitted direction to admit a refracted ray at that boundary, which condition on aa is necessary and sufficient? Grazing critical emergence is allowed.
Not attempted
Clampview / emission geometry; not to scale
A
a≤R/na\le R/n
B
a=0a=0
C
a≤R1−1/n2a\le R\sqrt{1-1/n^2}
D
a≥R/na\ge R/n
51
A point source lies 6 cm6\text{ cm} from the centre of a sphere of radius 10 cm10\text{ cm} and index 22 in air. A ray is emitted at angle γ\gamma to the outward radial direction through the source. Let γ0\gamma_0 be the smallest positive angle at which first-boundary emergence becomes grazing critical. Find 36sin⁡2γ036\sin^2\gamma_0. Give the numerical answer to two decimal places.
Not attempted
Clampview / emission geometry; not to scale
Your answer: —
Correct answer: 25
52
A horizontal medium has an index decreasing continuously downward from ntn_t to nbn_b, with 1<nb<nt1<n_b<n_t. A downward ray starts at the top with angle θt\theta_t from the vertical. In the layered-ray limit, which range of p=ntsin⁡θtp=n_t\sin\theta_t makes it turn before reaching the bottom, without any discrete reflecting surface?
Not attempted
A
0<p<nt−nb0<p<n_t-n_b
B
nb<p<ntn_b<p<n_t
C
p<nbp<n_b
D
p>ntp>n_t
53
A meridional ray traverses a straight core of index ncn_c with wall medium nw<ncn_w<n_c, entering through a flat end from index n0n_0. Let the outside launch angle be θ\theta, and take all wall encounters away from edges. Strict TIR is required. Select all correct statements.
Not attempted
A
Guiding requires n02sin⁡2θ<nc2−nw2n_0^2\sin^2\theta<n_c^2-n_w^2.
B
If nc2−nw2>n02n_c^2-n_w^2>n_0^2, all external launch angles below ninety degrees are guided.
C
Increasing n0n_0 always widens the external acceptance cone.
D
No strict wall TIR survives when nw=ncn_w=n_c.
54
A source is under two horizontal layers of thicknesses a,ba,b and indices n1,n2>1n_1,n_2>1, with air above. A top disk centred over the source blocks all escaping rays. The two thickness-index pairs are interchanged. What happens to the minimum required disk radius?
Not attempted
A
It is multiplied by n2/n1n_2/n_1.
B
It is unchanged only if a=ba=b.
C
It remains unchanged.
D
It is multiplied by n1/n2n_1/n_2.
55
A tiny luminous inclusion is on a diameter of a glass sphere of radius 12 cm12\text{ cm}, index 3/23/2, at 4 cm4\text{ cm} from the centre. An air observer views from the nearer end of that diameter. Find the apparent distance of the inclusion from the near surface in centimetres, using paraxial rays. Give the numerical answer to two decimal places.
Not attempted
Cinclusionair observerR = 12 cm; inclusion 4 cm from C
Your answer: —
Correct answer: 6.86
56
A point inclusion on a diameter of a sphere of radius RR and index n>1n>1 lies a distance aa from the centre, 0<a<R0<a<R. Air observers view separately from the near and far ends using paraxial rays. Assume R>(n−1)aR>(n-1)a. What is the ratio of near-view apparent depth to far-view apparent depth, each measured inward from its own surface?
Not attempted
Cinclusionview / emission geometry; not to scale
A
R−aR+a\dfrac{R-a}{R+a}
B
R+(n−1)aR−(n−1)a\dfrac{R+(n-1)a}{R-(n-1)a}
C
R−an(R+a)\dfrac{R-a}{n(R+a)}
D
(R−a)[R−(n−1)a](R+a)[R+(n−1)a]\dfrac{(R-a)[R-(n-1)a]}{(R+a)[R+(n-1)a]}
57
A spherical surface separates index n1n_1 on its incident side from n2>n1n_2>n_1 on the outgoing side, with R>0R>0. Its paraxial conjugates obey the Cartesian convention. Define Φ=(n2−n1)/R\Phi=(n_2-n_1)/R and let mm be signed transverse magnification. Select all correct statements.
Not attempted
A
dv/du=(n2/n1)m2dv/du=(n_2/n_1)m^2.
B
The magnitudes of front and rear focal distances are equal.
C
u=−2n1/Φu=-2n_1/\Phi gives transverse magnification −1-1.
D
dv/du=−m2dv/du=-m^2.
58
A long transparent rod of index 3/23/2 has a convex entrance face with radius +12 cm+12\text{ cm}, and a plane exit face 48 cm48\text{ cm} farther along the axis. The rod is in air. An axial point is 36 cm36\text{ cm} before the entrance face. Find the signed position of the final image from the exit face, positive downstream, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
object: −36 cmrod length: 48 cmconvex entranceplane exit
Your answer: —
Correct answer: 40
59
A glass sphere of radius RR and index nn is in air and receives an axial parallel paraxial beam. Assume 1<n<21<n<2. Where is its final focus relative to the rear surface?
Not attempted
A
R(2−n)2(n−1)\dfrac{R(2-n)}{2(n-1)} downstream.
B
nR2(n−1)\dfrac{nR}{2(n-1)} downstream.
C
Rn−1\dfrac{R}{n-1} downstream.
D
R(n−1)2−n\dfrac{R(n-1)}{2-n} upstream.
60
A thin lens has radii R1=+RR_1=+R, R2=−2RR_2=-2R and glass index 3/23/2. It is immersed in a medium of index nmn_m. All distances are Cartesian, and the same medium occupies both sides. A real object is at distance 4R4R in front. Select all correct statements.
Not attempted
A
In air the transverse magnification is −1/2-1/2.
B
In index 6/56/5 the real image lies 8R8R beyond the lens.
C
At nm=3/2n_m=3/2 the object has a finite new real image beyond the lens.
D
For nm>3/2n_m>3/2 this lens becomes diverging.
61
Two thin lenses in contact in air are made of the same material of index 3/23/2. Their shape factors (1/R1−1/R2)(1/R_1-1/R_2) are +1/10+1/10 and −1/20-1/20, in cm−1\text{cm}^{-1}. The pair is immersed in index 5/45/4. A real object is 150 cm150\text{ cm} in front. Find the final real image distance in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectcontact pair in liquid150 cm
Your answer: —
Correct answer: 300
62
A spherical entrance surface has n1=1,n2=3/2,R=+10 cmn_1=1,n_2=3/2,R=+10\text{ cm}. Match signed object coordinates in List I to signed image coordinates in centimetres in List II. List IList IIP: u=-40 cm1: −30Q: u=-10 cm2: 45/2R: u=+20 cm3: 60S: u=+60 cm4: 155: 30\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: u=-40 cm}&\text{1: }-30\\\text{Q: u=-10 cm}&\text{2: }45/2\\\text{R: u=+20 cm}&\text{3: }60\\\text{S: u=+60 cm}&\text{4: }15\\\text{}&\text{5: }30\end{array}
Not attempted
A
P-2, Q-3, R-1, S-4
B
P-3, Q-1, R-4, S-2
C
P-1, Q-4, R-2, S-3
D
P-4, Q-2, R-3, S-1
63
Two thin lenses made of the same glass of index ngn_g have shape factors K1,K2K_1,K_2 and fixed separation d>0d>0. They are immersed in a medium nmn_m. Write k=ng/nm−1k=n_g/n_m-1, and assume K1K2≠0K_1K_2\ne0. Apart from the index-matched case k=0k=0, at what kk does their equivalent power vanish?
Not attempted
A
k=dK1K2K1+K2k=\dfrac{dK_1K_2}{K_1+K_2}
B
k=K1−K2dK1K2k=\dfrac{K_1-K_2}{dK_1K_2}
C
k=K1+K2dK1K2k=\dfrac{K_1+K_2}{dK_1K_2}
D
k=−(K1+K2)k=-(K_1+K_2)
64
An equiconvex thin lens of focal length ff in air is cut into two plano-convex halves by a plane perpendicular to its axis. A different identical lens is cut along its axis into two semicircular pieces. Ignore thickness and edge diffraction; all pieces retain their original curved surfaces. Select all correct statements.
Not attempted
A
The two plano-convex halves separated by 4f4f are afocal.
B
Each semicircular axial-cut piece has focal length 2f2f.
C
The two plano-convex halves in contact restore focal length ff.
D
Each axial-cut piece can form the entire geometrical image if its aperture accepts rays from every object point.
65
A thin lens of index nn lies between external indices nLn_L and nRn_R, with signed radii R1,R2R_1,R_2 for left-to-right propagation. Define Φ=(n−nL)/R1+(nR−n)/R2\Phi=(n-n_L)/R_1+(n_R-n)/R_2. What is the signed transverse magnification for conjugates u,vu,v, with the surface separation neglected?
Not attempted
A
m=v/um=v/u
B
m=−v/um=-v/u
C
m=nRvnLum=\dfrac{n_Rv}{n_Lu}
D
m=nLvnRum=\dfrac{n_Lv}{n_Ru}
66
A thin lens separates air on the left from liquid of index 4/34/3 on the right. Its glass index is 3/23/2, and left-to-right radii are +12+12 and −12 cm-12\text{ cm}. A real object is 36 cm36\text{ cm} to its left. Find the signed transverse magnification. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlens: air | liquid36 cm
Your answer: —
Correct answer: -1
67
Two thin lenses of focal lengths +12+12 and −24 cm-24\text{ cm} are at x=0x=0 and x=6 cmx=6\text{ cm}. An object is at x=−24 cmx=-24\text{ cm}. Find the signed final image coordinate xx, in centimetres, for light travelling left to right. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectf₁ = 12 cmf₂ = −24 cm6 cm
Your answer: —
Correct answer: 78
68
Thin lenses of focal lengths f1,f2>0f_1,f_2>0 are separated by d=f1+f2d=f_1+f_2. A parallel paraxial ray bundle of axial radius aa enters the first lens. What is the signed ratio of output ray height to input ray height, and what is the output convergence? The option sketches illustrate f2/f1=2f_2/f_1=2; the captions specify the general signed ratio and output convergence.
Not attempted
A
Height ratio −f2/f1-f_2/f_1; parallel output.
referencesigned output beam radius
B
Height ratio −f1/f2-f_1/f_2; parallel output.
referencesigned output beam radius
C
Height ratio +1+1; parallel output.
referencesigned output beam radius
D
Height ratio −f2/f1-f_2/f_1; converging output at f2f_2.
referencesigned output beam radius
69
Two thin converging lenses each have focal length ff, separated by 2f2f. A small object lies 2f2f before the first lens, and the observation plane lies 2f2f after the second. Let the first-lens aperture radius be aa and the second be bb. Paraxial rays from an on-axis object point are considered. Select all correct statements. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectffscreen2f
A
The stated object does not have a finite sharp image in the stated observation plane.
B
Afocality alone guarantees magnification +1+1 for the stated conjugates.
C
For an object at ff before the first lens, a real image forms at ff after the second with magnification −1-1.
D
Every ray from the axial object at 2f2f reaches the second lens at height zero in the ideal model.
70
A thin converging lens of focal length 10 cm10\text{ cm} is at x=0x=0. A plane mirror is at x=15 cmx=15\text{ cm}. A small off-axis object is at x=−30 cmx=-30\text{ cm}. Light passes through the lens, reflects once and passes through the lens again. Find the final image distance to the left of the lens, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectf = 10 cmplane mirror15 cm
Your answer: —
Correct answer: 30
71
A lens of focal length f>0f>0 is followed at distance dd by a plane mirror. A point object is at the front focal plane of the lens. After a forward lens pass, one reflection, and a return lens pass, where does its final image form and with what transverse magnification? Mirror and lens are large enough to pass all considered paraxial rays. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlens fplane mirrord (schematic)
A
At distance f/2f/2 left of the lens, with m=+1m=+1.
B
At the original object plane, with m=−1m=-1, independent of dd.
C
At the original object plane, with m=+1m=+1, independent of dd.
D
At distance 2d+f2d+f left of the lens, with m=−1m=-1.
72
A lens of focal length 12 cm12\text{ cm} is at x=0x=0; a plane mirror is at x=18 cmx=18\text{ cm}. An object is at x=−36 cmx=-36\text{ cm}. The mirror is displaced right by an infinitesimal amount while lens and object stay fixed. Find the derivative of the final image coordinate xIx_I with respect to the mirror coordinate xMx_M at this setting. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectf = 12 cmmoving plane mirror18 cm at instant
Your answer: —
Correct answer: 8
73
A thin converging lens has focal length ff. A real object plane and a real screen plane are fixed a distance D>4fD>4f apart. The lens has two conjugate positions. At one, the magnitude of transverse magnification is r>1r>1; at the other it is 1/r1/r. Select all correct statements.
Not attempted
A
D/f=r+2+1/rD/f=r+2+1/r.
B
The two signed magnifications multiply to −1-1.
C
The lens-position separation is D(r−1)/(r+1)D(r-1)/(r+1).
D
The geometric mean of the two image-height magnitudes equals the object height.
74
A converging thin lens of focal length ff is translated transversely by aa from the axis, without tilt. A fixed object of height zero on the original axis is at distance s>fs>f before it. Where is the real image height relative to the original axis? Option sketches use s=2fs=2f and show image height for a positive lens shift; the formula captions define each general claim.
Not attempted
A
yI=afs−fy_I=\dfrac{af}{s-f}
referenceimage shift for a > 0, s = 2f
B
yI=−ass−fy_I=-\dfrac{as}{s-f}
referenceimage shift for a > 0, s = 2f
C
yI=ass−fy_I=\dfrac{as}{s-f}
referenceimage shift for a > 0, s = 2f
D
yI=ay_I=a
referenceimage shift for a > 0, s = 2f
75
Two thin lenses each have focal length 10 cm10\text{ cm} and are separated by 15 cm15\text{ cm}. Their centres have heights +1+1 and −1 cm-1\text{ cm} relative to a common horizontal axis; their planes are parallel. An object on the original axis is 20 cm20\text{ cm} before the first lens. Find the signed height of the final real image in centimetres. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 1
76
A thin converging lens is cut along its principal axis into upper and lower semicircular pieces, then the pieces are shifted to centres +a+a and −a-a without tilt. A point object on the old axis lies s>fs>f in front. Assume each piece receives a nonzero ray bundle from the object and ignore edge diffraction. What is the separation of the two real images?
Not attempted
A
2a2a
B
2afs−f\dfrac{2af}{s-f}
C
2ass−2f\dfrac{2as}{s-2f}
D
2ass−f\dfrac{2as}{s-f}
77
A thin diverging lens has focal length −f-f, f>0f>0. Its incoming beam would focus a distance u>0u>0 beyond the lens if the lens were removed. Use Cartesian signs for the transmitted image. Select all correct statements.
Not attempted
A
For u=f/2u=f/2, v=fv=f and m=2m=2.
B
For u=2fu=2f, v=−2fv=-2f and m=−1m=-1.
C
For u=fu=f, the output is collimated.
D
For every positive uu, the transmitted image is virtual.
78
A converging lens of focal length 15 cm15\text{ cm} forms an image on a fixed screen at 60 cm60\text{ cm} beyond it. A diverging lens of focal length −30 cm-30\text{ cm} is inserted 30 cm30\text{ cm} after the converging lens. The object is moved so the final image remains on the same screen. Find the new object distance before the first lens, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectf₁ = 15 cmf₂ = −30 cmscreen60 cm
Your answer: —
Correct answer: 22.5
79
A point object moves along the axis of a stationary converging lens. Its coordinate is u(t)=−f−f2/(vt+b)u(t)=-f-f^2/(vt+b), where v,b>0v,b>0 have units making vt+bvt+b a length. What is the transmitted image motion?
Not attempted
A
Uniform downstream speed vv.
B
Uniform upstream speed vv.
C
Constant downstream acceleration v2/fv^2/f.
D
A virtual image moving at speed vv.
80
Two lenses, first +f+f and second −f-f, have variable separation dd. For a parallel axial input beam, match dd in List I to signed rear focal distance measured from the second lens in List II; positive means downstream. List IList IIP: d=f/21: −f/2Q: d=f2: 0R: d=2f3: −2f/3S: d=3f4: f5: 2f\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: d=f/2}&\text{1: }-f/2\\\text{Q: d=f}&\text{2: }0\\\text{R: d=2f}&\text{3: }-2f/3\\\text{S: d=3f}&\text{4: }f\\\text{}&\text{5: }2f\end{array}
Not attempted
A
P-3, Q-4, R-2, S-1
B
P-4, Q-2, R-1, S-3
C
P-2, Q-1, R-3, S-4
D
P-1, Q-3, R-4, S-2
81
Two thin lenses of focal lengths +f+f and −2f-2f are separated by d>0d>0, with the converging one first. A parallel axial beam enters. Treat all lenses as having unlimited apertures. Select all correct statements.
Not attempted
A
The equivalent power is positive for every d>0d>0.
B
For every d>0d>0, the output has a real focus downstream of the second lens.
C
For d>fd>f, the output is diverging although the equivalent power is positive.
D
As d→0d\to0, the rear focal distance tends to 2f2f.
82
A thin lens of focal length 20 cm20\text{ cm} is at x=0x=0. A concave mirror facing left is at x=30 cmx=30\text{ cm} and has focal-length magnitude 15 cm15\text{ cm}. An object is at x=−40 cmx=-40\text{ cm}. Light passes through the lens, reflects, and returns through the lens. Find the final real image distance to the left of the lens, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectf = 20 cmmirror |f| = 15 cm30 cm
Your answer: —
Correct answer: 120
83
A lens of focal length ff is immediately in front of a concave mirror of focal-length magnitude g>0g>0, with their vertex planes coincident in the zero-separation model. A real object is at distance ss. The ray passes through the lens twice and reflects once. What positive effective focal-length magnitude makes this folded system equivalent to a concave mirror? The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlens / mirrorvertex separation neglected
A
F=fg2f+gF=\dfrac{fg}{2f+g}
B
F=f+gF=f+g
C
F=fgf+2gF=\dfrac{fg}{f+2g}
D
F=fgf+gF=\dfrac{fg}{f+g}
84
A lens of focal length 30 cm30\text{ cm} is immediately before a convex mirror of focal-length magnitude 60 cm60\text{ cm}, in the zero-separation model. An object is 40 cm40\text{ cm} in front. After the lens-mirror-lens sequence, find the real image distance in front, in centimetres. Give the numerical answer to two decimal places. The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectlens / mirrorvertex separation neglected
Your answer: —
Correct answer: 40
85
A small object has a tiny axial displacement dudu and tiny transverse displacement dhdh near an off-axis point (u,h)(u,h) of a fixed thin lens. Its image is (v,mh)(v,mh), where m=v/um=v/u and 1/v−1/u=1/f1/v-1/u=1/f. Displacements are infinitesimal and lens coordinates Cartesian. Select all correct statements.
Not attempted
A
dv=m2dudv=m^2du.
B
dyI=m dh−(hm2/f)dudy_I=m\,dh-(h m^2/f)du.
C
A purely axial object displacement always gives zero transverse image displacement.
D
For finite rods, using one endpoint value of m2m^2 always gives the exact axial image length.
86
A thin converging lens of focal length ff images a straight luminous line in its object half-plane. Using s=−u>fs=-u>f, the object line is h=As+Bh=As+B over a short interval that stays beyond the focus. What is the locus of the image in coordinates (v,H)(v,H)? The sketches illustrate the formula captions for A=B=f=1A=B=f=1; the requested answer is the general image locus.
Not attempted
A
H=Av+BH=Av+B
vHexample A = B = f = 1
B
H=−Av−BH=-Av-B
vHexample A = B = f = 1
C
H=−(A−B/f)v−BH=-(A-B/f)v-B
vHexample A = B = f = 1
D
H=−(A+B/f)v+BH=-(A+B/f)v+B
vHexample A = B = f = 1
87
A thin lens of focal length 10 cm10\text{ cm} images a luminous line whose coordinates satisfy h=s/10−1 cmh=s/10-1\text{ cm}, where ss is its distance in centimetres in front of the lens and 20≤s≤3020\le s\le30. Find the absolute difference in image heights between the two endpoints, in centimetres. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 0
88
A lens of focal length ff images an object at distance 2f2f onto a screen 2f2f beyond. A normal slab with shift Δ=t(1−1/n)\Delta=t(1-1/n) is inserted entirely before the screen. Object and screen remain fixed, but the lens may slide; both possible lens locations remain before the slab. For which positive Δ\Delta can a sharp image still be restored? The accompanying layout is schematic; distances and surface shapes are defined by the text.
Not attempted
objectmovable lensslabscreen4f
A
For no positive Δ\Delta.
B
For every 0<Δ<f0<\Delta<f.
C
Only for Δ=f\Delta=f.
D
For every 0<Δ<2f0<\Delta<2f.
89
A prism in air has apex angle AA and index n>1n>1. Two distinct non-grazing incidence angles i1,i2i_1,i_2 produce the same deviation δ\delta through the two refracting faces, with positive internal refraction angles. Which relation must hold?
Not attempted
Aincident raysecond faceprismschematic; angles given in text
A
i1i2=Aδi_1i_2=A\delta
B
i1+i2=A+δi_1+i_2=A+\delta
C
i1+i2=Ai_1+i_2=A
D
i1+i2=2Ai_1+i_2=2A
90
A prism has apex angle 60∘60^\circ. Two incidence angles 30∘30^\circ and 60∘60^\circ produce the same deviation through its two faces in air. Find 4n24n^2, where nn is the refractive index. Assume ordinary transmitted paths with positive internal angles. Give the numerical answer to two decimal places.
Not attempted
Aincident raysecond faceprismschematic; angles given in text
Your answer: —
Correct answer: 7.64
91
A prism of apex angle AA and index n>1n>1 in air is set at minimum deviation δm\delta_m. Take A<2sin⁡−1(1/n)A<2\sin^{-1}(1/n), so the setting is non-grazing. The prism is then used with its faces interchanged, without changing the magnitude of the incidence angle relative to the new entrance normal. Select all correct statements.
Not attempted
Aincident raysecond faceprismschematic; angles given in text
A
Face interchange preserves the minimum deviation setting.
B
n=sin⁡[(A+δm)/2]/sin⁡(A/2)n=\sin[(A+\delta_m)/2]/\sin(A/2).
C
Just above the minimum, the paired incidence angles sum to A+δA+\delta.
D
At minimum deviation, δm=(n−1)A\delta_m=(n-1)A exactly for every apex angle.
92
A prism has apex angle A=2CA=2C, where C=sin⁡−1(1/n)C=\sin^{-1}(1/n) for its index n>1n>1 in air. A ray is required to enter and emerge at angles strictly below 90∘90^\circ. What can be concluded about paths through its two refracting faces?
Not attempted
Aincident raysecond faceprismschematic; angles given in text
A
Every entered ray emerges non-grazingly.
B
There are two non-grazing paths for every deviation.
C
No required transmitted path exists; only a double-grazing boundary is possible.
D
Exactly one non-grazing symmetric path exists.
93
Two thin prisms are combined with opposing bases. Their mean indices are 3/23/2 and 5/35/3, and their violet-minus-red index differences are 1/1001/100 and 1/501/50. The first apex angle is 6∘6^\circ. Find the magnitude of the net mean deviation, in degrees, when the pair has zero violet-red angular dispersion. Use the first-order thin-prism approximation throughout. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 1
94
Two thin prisms in contact have signed mean deviations d1,d2d_1,d_2 and positive material dispersive powers ω1≠ω2\omega_1\ne\omega_2. Mean deviation and violet-red angular dispersion are D=d1+d2D=d_1+d_2 and S=ω1d1+ω2d2S=\omega_1d_1+\omega_2d_2. Neither prism is absent. Select all correct statements.
Not attempted
A
A nontrivial direct-vision pair has nonzero angular dispersion.
B
A nontrivial angularly achromatic pair has nonzero mean deviation.
C
Nontrivial simultaneous cancellation is possible by changing only the two apex angles.
D
Angular achromatism requires the two mean deviations to have opposite signs.
95
Two thin prisms at the same plane have total violet-red angular deviation difference zero but nonzero common mean deviation δ\delta, small in radians. They lie just before an ideal achromatic converging lens of focal length ff. A parallel beam along the lens axis falls on them. What happens in the rear focal plane?
Not attempted
A
Violet and red coincide on the original axis.
B
The two colours are separated by 2fδ2f\delta.
C
Neither colour has a real focus.
D
Violet and red coincide at height fδf\delta from the axis.
96
Two contact thin prisms have signed mean deviations d1,d2d_1,d_2 and dispersive powers ω,2ω\omega,2\omega, with ω>0\omega>0. Match each ratio d2/d1d_2/d_1 in List I to the pair (D/d1,S/(ωd1))(D/d_1,S/(\omega d_1)) in List II. Assume d1≠0d_1\ne0. List IList IIP: ratio -11: (1/2,0)Q: ratio -1/22: (3/2,2)R: ratio 03: (0,−1)S: ratio +1/24: (1,1)5: (2,3)\begin{array}{ll}\text{List I}&\text{List II}\\\text{P: ratio -1}&\text{1: }(1/2,0)\\\text{Q: ratio -1/2}&\text{2: }(3/2,2)\\\text{R: ratio 0}&\text{3: }(0,-1)\\\text{S: ratio +1/2}&\text{4: }(1,1)\\\text{}&\text{5: }(2,3)\end{array} Include the zero-apex limit for the second prism when its deviation is zero.
Not attempted
A
P-3, Q-1, R-4, S-2
B
P-1, Q-4, R-2, S-3
C
P-4, Q-2, R-3, S-1
D
P-2, Q-3, R-1, S-4
97
A parallel beam passes two thin prisms separated by 50 cm50\text{ cm}. Their signed violet-minus-red angular deviations are +0.004+0.004 and −0.004-0.004 radians. Initially both colour rays coincide at the first prism; prism thickness and lateral shift at each prism are negligible. Find their absolute lateral separation in millimetres at a screen any distance after the second prism. Work to first order in angles. Give the numerical answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 2
98
A parallel slab and a prism are both in air and have normal dispersion, with violet index larger than red. The slab has parallel faces; the prism is non-parallel and admits the stated transmitted rays. Ignore diffraction and absorption. Select all correct statements.
Not attempted
A
A slab can separate the two colour rays laterally without separating their outgoing directions.
B
Violet has larger minimum deviation than red for the same prism apex angle in the transmitting regime.
C
The two colours necessarily have the same minimum-setting incidence angle.
D
Zero net slab angular deviation implies zero colour-dependent lateral displacement.
99
A prism of apex angle AA is immersed in a liquid. Its absolute index is ngn_g and the liquid index is nl<ngn_l<n_g. A two-face non-grazing transmitted path exists for some incidence angle. What inequality does this impose?
Not attempted
Aincident raysecond faceprismschematic; angles given in text
A
ngnl<1/sin⁡A\dfrac{n_g}{n_l}<1/\sin A
B
ngnl<1sin⁡(A/2)\dfrac{n_g}{n_l}<\dfrac{1}{\sin(A/2)}
C
ng<1/sin⁡(A/2)n_g<1/\sin(A/2)
D
ngnl>1sin⁡(A/2)\dfrac{n_g}{n_l}>\dfrac{1}{\sin(A/2)}
100
Two thin lenses in contact have mean powers P1,P2P_1,P_2 and positive dispersive powers ω1,ω2\omega_1,\omega_2, defined here by ΔPj=ωjPj\Delta P_j=\omega_jP_j for violet-minus-red power differences. Two contact thin prisms have mean deviations d1,d2d_1,d_2 with the same two dispersive powers. Assume ω1≠ω2\omega_1\ne\omega_2, and nonzero first-element power and deviation. Select all correct statements.
Not attempted
A
The required ratios P2/P1P_2/P_1 and d2/d1d_2/d_1 are equal.
B
The achromatic lens combination must have zero mean power.
C
For ω2>ω1\omega_2>\omega_1 and P1>0P_1>0, the achromatic pair has positive mean power.
D
The stated two-colour condition guarantees equal behaviour for every wavelength.
The same signed linear cancellation controls lens powers and prism deviations, but the physical outputs differ.
Lens achromatism requires ω1P1+ω2P2=0\omega_1P_1+\omega_2P_2=0, hence P2/P1=−ω1/ω2P_2/P_1=-\omega_1/\omega_2. Prism angular achromatism similarly requires d2/d1=−ω1/ω2d_2/d_1=-\omega_1/\omega_2. The remaining lens power and prism deviation are respectively P1(1−ω1/ω2)P_1(1-\omega_1/\omega_2) and d1(1−ω1/ω2)d_1(1-\omega_1/\omega_2), both nonzero. These are two-colour cancellation conditions, not proof of zero variation at every wavelength. Equal positive powers cannot cancel positive-dispersive colour-power changes.