Indian National Physics Olympiad (INPhO)-2023

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research
V. N. Purav Marg, Mankhurd, Mumbai, 400 088

Question Paper

Date: 29 January 2023

Time: 09:00-12:00 (3 hours) Maximum Marks: 60
Instructions Roll No.:

1. This booklet consists of 6 pages and total of 5 questions. Write roll number at the top wherever
asked.

2. Booklet to write the answers is provided separately. Instructions to write the answers are on the
Answer Booklet.

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3. Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calcula-
tors.
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4. Please submit the Answer Sheet at the end of the examination. You may retain the
Question Paper.

Table of Constants
Speed of light in vacuum c 3.00 × 108 m·s−1
6.63 × 10−34 J·s
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Planck’s constant h
–
h h/2π
−2
Universal constant of Gravitation G 6.67 × 10−11 N·m2 ·kg
Magnitude of electron charge e 1.60 × 10−19 C
Rest mass of electron me 9.11 × 10−31 kg
Value of 1/4π0 9.00 × 109 N·m2 ·C−2
Avogadro’s number NA 6.022 ×1023 mol−1
Acceleration due to gravity g 9.81 m·s−2
Universal Gas Constant R 8.31 J· K−1 ·mol−1
R 0.0821 l·atm·mol−1 ·K−1
Boltzmann constant KB 1.3806 × 10−23 J·K−1
Permeability constant µ0 4π × 10−7 H·m−1
1 Angstrom unit 1Å 1 × 10−10 m
1 micro unit 1µ 1 × 10−6 units
1 electron volt 1 eV 1.6 × 10−19 J

Q No 1 2 3 4 5 Total

Maximum Marks 6 6 16 16 16 60
Page 2 INPhO 2023 Questions Last four digits of Roll No.:

1. [6 marks] Dancing on the floor

There are various apps that record the intensity of an audio signal. An app (WaveEditorTM here)
displays the audio signal as a wave, whose amplitude is proportional to the audio signal’s loudness.
A smartphone with this app recording the sound signal is kept on a uniformly built flat floor of a
classroom.

13.832
12.848

14.000

11.409
13.284
10.260 12.244
13.602

A perfectly small spherical steel ball is thrown up such that it almost touches the ceiling and comes
back without hitting. The ball hits the floor and thereafter it keeps bouncing. The app records the

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sound signal produced when the ball hits the floor on every bounce. A screenshot of the recording
is shown. The timestamps (in seconds) of the first eight consecutive bounces are also shown next
to the peak. For example, the app records a peak at 10.260 s when the first time the ball hits the
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floor.
Make reasonable assumptions, when the ball hits the floor and calculate the height of the classroom
from the given data. State your assumptions clearly.
2. Knock it off!
Consider a 100 W small isotropic source of blue light of wavelength 4500Å. A metallic surface of
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1.00 cm2 and work function 2.20 eV is kept at a distance of 1.00 m from the source and oriented to
receive normal radiation.
(a) [2 marks] Assume that all the energy is uniformly absorbed by atoms on the top layer of the
surface. Also, all the energy absorbed by an atom on the surface is taken up by one electron.
The radius of the atom is 1.00Å. Estimate the time τe needed by the electron to receive 1.00
eV of energy.
(b) [1 marks] According to the above classical model, how many electrons are emitted by the
metallic surface in time τe ?

(c) [2 marks] In quantum theory, photons are emitted and absorbed as quanta. Assuming
photoelectric efficiency of 1%, calculate the rate of emission of electrons (Ne ) from the surface.
(d) [1 marks] Assuming further that all the emitted photoelectrons move normal to the surface
what would be the maximum current density (Jmax ) one may expect?
3. [16 marks] Work in progress

One mole of an ideal monoatomic gas goes through a linear process from P
A to B as shown in the pressure-volume (P -V ) diagram. The temperature A
at A is TA = 227◦ C. The process is such that, the temperature decreases
and the heat is continuously supplied to the gas. The ratio of the
specific heat at the constant pressure to that at the constant volume is
B
5/3. Obtain the expression for the maximum work (Wmax ) the gas can
perform in such a process. Calculate Wmax .
V
Page 3 INPhO 2023 Questions

4. Electrostatic TikTok
Consider a fixed infinite vertical thin rod (shown by the red color in the figure below) of linear
charge density λ along the z-axis at the origin (see figure below). A uniformly charged ring of total
charge Q, mass M , and radius a is placed with its center at the origin in the x-y plane. Point P
is an arbitrary point on the ring. The projection of point P on x-y plane makes an angle θ with
respect to the x-axis in the anticlockwise direction as seen from the top.
The ring is now given an initial angular velocity ω0 about the x-axis. We define the angle α which
the plane of the ring makes with the x-y plane. This is illustrated by drawing line segment AB in
the plane of the ring. Initially α = 0. Ignore gravity.

P B
α
y
θ
A

You may find the following differentiation useful

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d 1
D= [tan−1 (q tan θ)] = [q(sec2 θ)] (4.1)
dθ 1 + (q tan θ)2

~ 0 ) due to the infinite rod at a point on

(a) [1 marks] State an expression for the electric field (E
the ring when α=0 in terms of x, y and θ, and related quantities.
(b) [2 marks] At some instant the ring makes an angle α. Derive an expression for the electric
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~ due to the infinite rod at a point on the ring in terms of θ, and α.

field E
(c) [1 marks] Find the net force F~ acting on the ring.
(d) [5 marks] Find the net torque ~τ acting on the ring in terms of α and the constants only.
Qualitatively plot torque as a function of α.
(e) [2 marks] Let the ring is in equilibrium with respect to α = 0. Derive an expression for
the time period T of small oscillations of the ring in terms of λ, and Q. Take λ = 0.1µ C/m,
Q = 2.0 µC, M = 50.0g, radius a = 5.0 cm, and ω0 = 1.0 rad/s. Calculate T .
(f) [2.5 marks] Find an expression for the potential energy U of the ring in terms of α. Quali-
tatively plot U as a function of α. Take the zero of potential energy to be at α = 0.
(g) [2.5 marks] Obtain the expression of maximum value of α (αmax ) in terms of ω0 . Calculate
αmax .
Page 4 INPhO 2023 Questions Last four digits of Roll No.:

5. If Prof. Snell had a smartphone

A typical smartphone screen is made up of mainly two
components: a sheet of touch-sensitive glass (where you air
move your finger to operate the phone) of thickness t at .
t touch-sensitive glass
the top and a LCD screen below it consisting of a regular
array of “RGB elements” that emit light. These elements h d air
have a separation of d between them. There is a thin air
gap of depth h between the touch-sensitive glass and the LCD RGB elements
LCD screen (see Fig. (1)for a cross sectional view). We Figure 1
estimate the value of h from the following experiment.
We use two smartphones (S-I and S-II) in this exercise – S-I is the target instrument in which we
want to estimate h, and S-II is the measuring instrument that can capture photos of the screen of
S-I which we then analyse using a image-processing software.
A digital image captured by the camera of a smartphone (S-II here) consists of discrete picture
elements called pixels. The image captured by S-II is processed through a software. A red color
reference line is drawn on the image (see Fig. 3(a)). The software plots the “brightness value” at
every point of the reference line as a function of the number of pixels from the left end of the line.
Thus, pixel number is a marker for distance here. First, we need to calibrate distance in terms of
pixel number.
The phone S-I is kept horizontal and the display is kept ON. A ruler is placed on its screen. S-II is
fixed above S-I to capture images. The image of the screen captured is shown in Fig (2).

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Figure 2

Figure 3(a) shows a part of the image of the ruler and its brightness value profile along the red
reference line in Fig. 3(b).

150
Brightness value

100

50

0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480
Pixel number
(a) (b)
Figure 3
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(a) [2 marks] State the number of pixels used by the camera of S-II to capture one centimeter
of the screen of S-I.
(b) [5 marks] We keep the setup the same as the last part. Next, a few small water drops are
placed on the glass screen of S-I beside the ruler (see Figs. 4(a) and 4(b) for a top and side
view, respectively). We model every drop as a hemispherical lens of radius R that magnifies
the array of RGB elements of the LCD screen of S-I (see Fig. 4(c); the figure is not to scale).

(b)

R air
.
t touch-sensitive glass

h d air

LCD RGB elements

(c)

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(a) (d)
Figure 4

Figure 4(d) shows the magnified image of the array of the RGB elements of the screen as
viewed from the top through one of the drops. This image is captured by S-II keeping the
camera settings and distance same as in the previous part. The brightness value profiles of
the images of the five chosen drops along the reference lines are shown in Fig. (5) on the next
page.
Using the profile plots, write the radius of the water drop (R in mm) and the corresponding
magnification (M ) of the separation d between the array of RGB elements of S-I for each
waterdrop lens. Use the table in the Summary Answer sheet to report your data. Describe
the method you have used and the calculations in the Detailed Answer sheet.
(c) [9 marks] For the given smartphone, t = 0.50 mm, the refractive indices of the touch-sensitive
glass, water drop, and the air to be 3/2, 4/3, and 1 respectively. Using the data table of the
previous part, plot a suitable linear graph to obtain the distance (h) of the RGB elements
from the touch-sensitive glass. Use the table given in the summary answer sheet to enter
the data used to plot the graph. Show your detailed theoretical calculation in the Detailed
Answer sheet.
Page 6 INPhO 2023 Questions Last four digits of Roll No.:
Water drop 1

Brightness value
200

150

100
Water drop 1

0 20 40 60 80 100 120 140 160 180 200 220

Pixel number
Water drop 2
Brightness value

200

150

100
Water drop 2
0 20 40 60 80 100 120 140 160 180 200 220
Pixel number)
200 Water drop 3
Brightness value

150

100

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Water drop 3 0 40 80 120 160 200 240 280 320
Pixel number

200 Water drop 4

Brightness value

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150

100

50

Water drop 4 0 40 80 120 160 200 240 280 320 360

Pixel number

200 Water drop 5

Brightness value

150

100

50

Water drop 5 0 40 80 120 160

Pixel number

Figure 5

**** END OF THE QUESTION PAPER ****