INPhO 2020, Теория — Задача 2. Charged cylindrical shell

Indian National Physics Olympiad – 2020 · 2 февруари 2020 г. · 12 т.

Внимание: Бележка към темата

The complete retained question booklet includes the summary answer boxes, sixteen blank detailed-answer pages and two blank graph sheets. All matching archived HBCSE key content is retained, including its handwritten graph and comment not for grading purposes. Printed sign/symbol discrepancies and the key cover page-count mismatch remain literal with labelled source notes.

Native cover and instructions

Indian National Physics Olympiad – 2020

Date: 02 February 2020. Time: 09:00–12:00 (3 hours). Maximum Marks: 80.

Roll Number: 2 0 □ □−□ □ □ □−□ □ □ □2\,0\,\square\,\square-\square\,\square\,\square\,\square-\square\,\square\,\square\,\square. Extra sheets attached: [blank box]. INO Centre (e.g. Ranchi): [blank line].

(Do not write below this line)

  1. This booklet consists of 23 pages (excluding this page) and total of 5 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 (maximum two) 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

ConstantSymbolValue
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}
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}}
RR0.0821 l⋅atm⋅mol−1⋅K−10.0821\ \mathrm{l\cdot atm\cdot mol^{-1}\cdot K^{-1}}
Permeability constantμ0\mu_04π×10−7 H⋅m−14\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}
QuestionMarksScore
113
212
315
420
520
Total80

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research

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

Native key notice

Please note that alternate/equivalent methods and different way of expressing final solutions may exist.

Source note: The retained key cover says “16 pages (excluding this page)”; the native PDF contains eighteen pages, including its appended handwritten graph. Its cover otherwise repeats the question cover instructions and constants.

Условие

An insulating uniformly charged cylindrical shell of radius aa lies with its axis along the zz axis. The shell’s moment of inertia per unit length about the zz axis and the surface charge density are II and σ\sigma respectively. The cylinder is placed in an external uniform magnetic field Bexz^B_{\mathrm{ex}}\hat z, and is initially at rest. Starting at t=0t=0 the external magnetic field is slowly reduced to zero. What is the final angular velocity ω\omega of the cylinder?

Complete charged cylinder with dashed z axis, both external field arrows, Bex z-hat and diameter2a.

Summary answer box: ω=\omega= [blank].

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

Решение

Оригинал в Архива: INPHO2020-Questions-en.pdf · решение от архива: INPHO2020-Solutions-20200220.pdf