Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Wang Jinhui

Chapter 1 Geometrical Optics - all with Video Answers

Educators

Chapter Questions

Problem 1

We model an optical fiber as a cylinder with refractive index $n_f$ surrounded by a cladding of refractive index $n_c<n_f$. The two circular ends of the optical fiber are not covered with cladding. Let $\theta$ be the angle of incidence of a light ray impinging at one of the ends of the fibre. Assuming that the surrounding medium has refractive index 1 , determine the range of $\theta$ for which the light ray is trapped in the optical fiber (i.e. cannot be transmitted to the cladding).

Prabhu Ramji

Numerade Educator

01:24

Problem 2

Field of View ${ }^*$
You are at an aquarium with a porthole of radius $R$, negligible thickness and refractive index $n=\frac{3}{2}$ embedded in an opaque ground. To observe aquatic lifeforms swimming at the bottom of the aquarium (in water with refractive index $\frac{4}{3}$ ) which is a distance $h$ below the ground, you peek through the porthole. What is the maximum area of the bottom that you can see?

Himanshu Kushwaha

Numerade Educator

00:30

Problem 3

Skewed Mirrors*
Consider two semi-infinite plane mirrors with their finite ends placed together. The angle subtended by the two mirrors is $\alpha$. An emitter is placed at point $\mathrm{P}$ in this two-dimensional plane. Furthermore, a receiver in the form of a circular arc of radius $r$ is sandwiched between the two mirrors. Find the largest angle $\theta$ at which a light ray is emitted from $P$ will eventually reach the receiver. How many reflections does this take?
(GRAPH CAN'T COPY)

Keshav Singh

Numerade Educator

02:58

Problem 4

Consider two semi-infinite plane mirrors with their finite ends placed together. The angle subtended by the two mirrors is $2 \alpha<\frac{\pi}{2}$. An emitter is placed at point $\mathrm{P}$ in this two-dimensional plane. If the perpendicular to the mirror from $P$ is of length $a$, determine the angle $\theta$ at which a ray can be emitted such that it returns to $P$ after a reflection from the top mirror, followed by a reflection by the bottom mirror. What is the distance travelled by the light ray between its emission from and return to point $P$ ?
(GRAPH CAN'T COPY)

Shoukat Ali

Other Schools

03:15

Problem 5

Consider two semi-infinite plane mirrors with their finite ends placed together. The angle subtended by the two mirrors is $\alpha$. A ray, that is parallel to mirror 2 , is incident on mirror 1 and after 8 reflections, it emerges parallel to mirror 1 (after a final reflection from mirror 2). Determine $\alpha$.

Sanat Mukherjee

Numerade Educator

03:46

Problem 6

Half of the surface of a glass sphere with refractive index $n$ is coated with silver. Determine the angle of deflection of a ray that impinges on the noncoated surface of the equator, at an angle of incidence $i$, after it exits the sphere. Under what conditions can a bundle of parallel light rays - incident on the non-coated half of the equator at small angles of incidence - emerge from the ball, still in a parallel bundle?

Vishal Gupta

Numerade Educator

02:24

Problem 7

A medium with a refractive index $n(y)$ fills the region $y>0$. A light ray traveling along the $\mathrm{x}$-direction in air strikes the medium at a right incidence angle at the origin and begins to propagate within the medium. Determine $n(y)$ if (a) the ray moves in a circular arc of radius $R$ and (b) the ray moves in a complete sinusoidal curve of amplitude $1 \mathrm{~m}$ and "wavelength" $\lambda$. Given that the largest refractive index is that of diamond with $n=2.5$ approximately, determine the maximum angular size of the circular arc and the minimum "wavelength" of the sinusoidal curve.

Keshav Singh

Numerade Educator

View

Problem 8

A light ray starts from the interior of the circumference of a solid circle of radius $R$ at $\frac{\pi}{4}$ radians with respect to the radial direction, radially inwards. Set the origin to be at the center of the circle. If the refractive index of the circle varies according to the relationship $n(r)=\sqrt{\frac{R^2}{r^2}+1}$, determine the magnitude of the angular displacement of the light ray when it reaches the center of the circle.

Lainey Roebuck

Numerade Educator

01:09

Problem 9

On a sweltering afternoon, a man walks along a road. The refractive index of air above the road obeys $n(y)=n_0(1+\alpha y)$ where $\alpha$ is a constant and $y$ is the height above the road. Firstly, explain qualitatively the reason behind this variation in refractive index and whether $\alpha$ is positive or negative. As a result of this refractive index gradient, the man cannot see the road beyond a certain distance $L$. If his eyes are a height $h$ above the ground, determine $L$. Finding the trajectory of a light ray emanating from the road would be a bonus.

Mishal Gul

Numerade Educator

03:30

Problem 10

An isotropic point source is placed at the center of a cube of edge length $l$ and refractive index $n>1$. If the medium surrounding the cube is vacuum and $\sin ^{-1} \frac{1}{n} \leq \frac{\pi}{4}$ radians, determine the minimum surface area on the cube that needs to be covered with opaque paint so that no light escapes the cube. Next, for all $n>1$, determine the minimum painted area if the paint is now perfectly reflective.

Claire Hammond

Numerade Educator

07:02

Problem 11

A room takes the shape of a right-angled isosceles triangle with base length $8 a$. The walls of the room are covered with mirrors and a square receiver of side length $a$ is placed at the right-angled corner of the room. A light ray is emitted at an infinitesimal distance away from the mid-point of the hypotenuse, at an angle $\theta$ with respect to the horizontal, such that $\cot \theta=8$. Determine the distance covered by the light ray before it impinges on the receiver.
(GRAPH CAN'T COPY)

Aatish Gupta

Numerade Educator

02:31

Problem 12

Suppose that you wish to cross from one side of a lake (point P) to the other side (point Q). The lake takes the form of a rectangular strip of width $d$ and the vertical distance $y$ between $\mathrm{P}$ and $\mathrm{Q}$ is much smaller than the horizontal distance $L$ between them. If you move at speed $v_1$ on shore and $v_2<v_1$ in water, draw the path that takes the least time from points $\mathrm{P}$ to $\mathrm{Q}$. You do not need to calculate the time taken along this path.

Narayan Hari

Numerade Educator

01:52

Problem 13

Derive the Lensmaker's formula (Eq. (1.19)) for a thin lens with refractive index $n$ and comprising radii of curvature $R_1$ and $R_2$ by considering two rays and applying Fermat's principle. This approach is more direct than that presented in this chapter.

Mayukh Banik

Numerade Educator

04:19

Problem 14

The atmosphere of the Earth can be modeled as an ideal gas with a uniform temperature $T$ and average mass $M$, wrapped around a uniform spherical Earth of radius $r_0$. The gravitational field strength in the region of the atmosphere can be taken to be that at the surface of the Earth, $g$. If the refractive index of a point in the atmosphere is proportional to the density at that point, $n=\alpha \rho$, determine the height $h$ above the surface of the Earth at which a light ray travels in a circle around the Earth. Hint: The ideal gas law is $p V=n R T$ where $p, V, n$ and $T$ are the pressure, volume, moles and temperature of the gas respectively while $R$ is the ideal gas constant.

Averell Hause

Carnegie Mellon University

01:07

Problem 15

A more rigorous derivation of the laws of reflection and refraction from Fermat's principle uses the Lagrangian formulation. The $O P L$ between two points $y\left(x_1\right)$ and $y\left(x_2\right)$ visited by a light ray is given by
$$
O P L=\int_{x_1}^{x_2} n(x, y) \sqrt{1+y^2} d x
$$
where $n(x, y)$ is the refractive index of the medium and $y(x)$ is the trajectory of the ray between the two fixed endpoints. Use your knowledge of the Lagrangian method to prove the following. Firstly, the path taken by a ray in a homogeneous medium is a straight line. Next, prove the laws of reflection and refraction (hint: under a suitable choice of coordinates, the Hamiltonian is conserved).

Robert Leedy

Numerade Educator

02:27

Problem 16

Determine the minimum distance between a real object and its image produced by a thin converging lens in terms of its focal length $f$. Ignore the unrealistic case where the object distance is 0 .

Maria Gordon

Numerade Educator

02:31

Problem 17

A wire with negligible thickness is placed a distance $u$ in front of a converging lens of unknown focal length and diameter $D$. When a screen is placed at a distance $L$ behind the lens, a smudge with an appreciable thickness $d$ is formed. Determine the possible focal lengths of the lens.

Mayukh Banik

Numerade Educator

04:12

Problem 18

The flat surfaces of two thin plano-convex lenses of common radius $R$ but different refractive indices $n_1$ and $n_2$ are glued together to form a thin converging lens. Determine the focal length of this lens.

Mirza Aslam Beig

Numerade Educator

04:53

Problem 19

A small ball is placed along the axis of a concave mirror of focal length $f$ at an object distance $u$. The concave surface of the mirror is filled with a thin layer of liquid of refractive index $n$. If the image of the ball is formed by the rays impinging on the mirror near its vertex, determine the location of the image.

Prabhat Tyagi

Numerade Educator

01:45

Problem 20

A glass prism in the shape of a quarter-cylinder rests on a horizontal table. A uniform, horizontal bundle of light impinges perpendicularly on its vertical plane surface as shown in the figure below. Note that all rays are above the surface of the table (though some are infinitesimally close to it). If the radius of the cylinder is $R=5 \mathrm{~cm}$ and the refractive index of glass is $n=1.5$, where on the table beyond the cylinder, will a patch of light be found? A range should be given.
(GRAPH CAN'T COPY)

Christopher Dzorkpata

Numerade Educator

02:56

Problem 21

Two converging lenses of focal lengths $f_1$ and $f_2$ are situated between an object and a screen, with the lens with focal length $f_1$ closer to the object. If we require an image to be produced on a screen which is at a distance $l$ away from the object with $l<2 f_1+4 f_2$, show that for a given object distance to the first lens, $u>f_1$, there is only one possible position for the second lens.

Ajay Singhal

Numerade Educator

02:36

Problem 22

An ant lies along the principal axis of a concave mirror of focal length $f$. If the ant begins moving, under what conditions will the velocities of the ant and its image be identical? Supposing that the ant travels such that the object distance $u$ increases at the rate $\frac{d u}{d t}=\frac{\alpha}{v-f}$ where $v$ is the object distance and $\alpha$ is a constant, starting from an initial object distance $u_0$, determine $v(t)$.

Aman Gupta

Numerade Educator

01:03

Problem 23

A right-angled isosceles prism of side length $9 \mathrm{~cm}$ and refractive index 1.5 is placed $6 \mathrm{~cm}$ away from a converging lens of focal length $f_1=20 \mathrm{~cm}$, followed by a diverging lens of focal length $f_2=-10 \mathrm{~cm}$ a distance $7 \mathrm{~cm}$ behind it. A $1 \mathrm{~cm}$ stick is located $8 \mathrm{~cm}$ above the prism, with one end aligned with the mid-point of the hypotenuse as shown in the figure below. Describe the final image of the stick and its magnification.
(GRAPH CAN'T COPY)

Dominador Tan

Numerade Educator