INPhO 2018, Теория — Задача 1. The Yukawa potential

Indian National Physics Olympiad – 2018 · 28 януари 2018 г. · 4 т.

Внимание: Бележка към темата

All seven matching archived keys are retained, including alternative ladder methods, unequal-resistor generalization, accepted ranges and the full handwritten graph. Printed refraction, hourglass, Fibonacci and notation inconsistencies remain literal with separate source notes. The full original includes nineteen blank detailed-answer sheets and two graph papers.

Native cover and instructions

Indian National Physics Olympiad – 2018

Date: 28 January 2018. Time: 09:00–12:00 (3 hours). Maximum Marks: 75.

Roll Number: 1 8 □ □−□ □ □ □−□ □ □ □1\,8\,\square\,\square-\square\,\square\,\square\,\square-\square\,\square\,\square\,\square.

I permit/do not permit (strike out one) HBCSE to reveal my academic performance and personal details to a third party.

Besides the International Physics Olympiad (IPhO) 2018, do you also want to be considered for the Asian Physics Olympiad (APhO) 2018? For APhO 2018 and its pre-departure training, your presence will be required in Delhi and Vietnam from April 28 to May 15, 2018. In principle, you can participate in both Olympiads. Yes/No.

Full Name (BLOCK letters) Ms./Mr.: [blank].

Extra sheets attached: [blank box]. Date: [blank]. Centre (e.g. Kochi): [blank]. Signature: [blank].

(Do not write below this line)

Instructions

  1. This booklet consists of 27 pages (excluding this sheet) and total of 7 questions.
  2. This booklet is divided in two parts: Questions with Summary Answer Sheet and Detailed Answer Sheet. Write roll number at the top wherever asked.
  3. The final answer to each sub-question should be neatly written in the box provided below each sub-question in the Questions & Summary Answer Sheet.
  4. You are also required to show your detailed work for each question in a reasonably neat and coherent way in the Detailed Answer Sheet. You must write the relevant Question Number(s) on each of these pages.
  5. Marks will be awarded on the basis of what you write on both the Summary Answer Sheet and the Detailed Answer Sheet. Simple short answers and plots may be directly entered in the Summary Answer Sheet. Marks may be deducted for absence of detailed work in questions involving longer calculations. Strike out any rough work that you do not want to be considered for evaluation.
  6. Adequate space has been provided in the answersheet for you to write/calculate your answers. In case you need extra space to write, you may request additional blank sheets from the invigilator. Write your roll number on the extra sheets and get them attached to your answersheet and indicate number of extra sheets attached at the top of this page.
  7. Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
  8. Use blue or black pen to write answers. Pencil may be used for diagrams/graphs/sketches.
  9. This entire booklet must be returned at the end of the examination.

Table of Constants

QuantitySymbolValue
Speed of light in vacuumcc3.00×108 m⋅s−13.00\times10^8\ \mathrm{m\cdot s^{-1}}
Planck’s constanthh6.63×10−34 J⋅s6.63\times10^{-34}\ \mathrm{J\cdot s}
ℏ\hbarh/2πh/2\pi
Universal constant of GravitationGG6.67×10−11 N⋅m2⋅kg−26.67\times10^{-11}\ \mathrm{N\cdot m^2\cdot kg^{-2}}
Magnitude of electron chargeee1.60×10−19 C1.60\times10^{-19}\ \mathrm C
Rest mass of electronmem_e9.11×10−31 kg9.11\times10^{-31}\ \mathrm{kg}
Rest mass of protonmpm_p1.67×10−27 kg1.67\times10^{-27}\ \mathrm{kg}
Value of 1/4πϵ01/4\pi\epsilon_09.00×109 N⋅m2⋅C−29.00\times10^9\ \mathrm{N\cdot m^2\cdot C^{-2}}
Avogadro’s numberNAN_A6.022×1023 mol−16.022\times10^{23}\ \mathrm{mol^{-1}}
Acceleration due to gravitygg9.81 m⋅s−29.81\ \mathrm{m\cdot s^{-2}}
Universal Gas ConstantRR8.31 J⋅K−1⋅mol−18.31\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}
Molar mass of water18.02 g⋅mol−118.02\ \mathrm{g\cdot mol^{-1}}
Permeability constantμ0\mu_04π×10−7 H⋅m−14\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}
QuestionMarksScore
14
28
312
49
510
617
715
Total75

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research

V. N. Purav Marg, Mankhurd, Mumbai, 400 088

Native key cover and marking policy

Solutions of Indian National Physics Olympiad – 2018

The key cover repeats the date, roll-number form, APhO participation question, nine instructions, thirteen constant rows, seven-question score table and HBCSE address of the question cover, but prints “16 pages (excluding this sheet)”. The retained key has eighteen PDF pages: the cover, sixteen numbered pages and one appended handwritten graph sheet.

On its next page the key states: ANY ALTERNATIVE METHOD OF SOLUTION TO ANY QUESTION THAT IS SCIENTIFICALLY AND MATHEMATICALLY CORRECT, AND LEADS TO THE SAME ANSWER WILL BE ACCEPTED WITH FULL CREDIT. PARTIALLY CORRECT ANSWERS WILL GAIN PARTIAL CREDIT.

Условие

In a nucleus, the attractive central potential which binds the proton and the neutron is called the Yukawa potential. The associated potential energy U(r)U(r) is

U(r)=−αe−r/λrU(r)=-\alpha\frac{e^{-r/\lambda}}r

Here λ=1.431 fm\lambda=1.431\ \mathrm{fm} (fm=10−15 m\mathrm{fm}=10^{-15}\ \mathrm m), rr is the distance between nucleons, and α=86.55 MeV⋅fm\alpha=86.55\ \mathrm{MeV\cdot fm} is the nuclear force constant (1 MeV=1.60×10−13 J1\ \mathrm{MeV}=1.60\times10^{-13}\ \mathrm J). Assume nuclear force constant α\alpha to be AℏcA\hbar c. Here ℏ=h/2π\hbar=h/2\pi and hh is Planck’s constant. In order to compare the nuclear force to other fundamental forces of nature within the nucleus of Deuterium ( 2H\,{}^2\mathrm H), let the constants associated with the electrostatic force and the gravitational force to be equal to BℏcB\hbar c and CℏcC\hbar c respectively. Here AA, BB and CC are dimensionless. State the expression and numerical values of AA, BB and CC.

Summary answer fields: A=A= [blank]; Value of A=A= [blank]; B=B= [blank]; Value of B=B= [blank]; C=C= [blank]; Value of C=C= [blank].

Detailed answers can be found on page numbers: [blank].

Native answer booklet: PDF pages8–25 and28 (printed pages7–24 and27) are nineteen blank Detailed Answers pages with a “Que. No.” box and an alternate-page “Last four digits of Roll No.:” field. Every sheet remains in the complete original; the first complete blank template is retained here.

Complete first native blank Detailed Answers sheet, printed page7, question-number box and whole answer frame.

Решение

Оригинал в Архива: INPhO2018-Question.pdf · решение от архива: INPhO2018-Solution-20180213.pdf