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Module Code: PHYS237001 Module Title: Maths 3 c© UNIVERSITY OF LEEDS Resit Mid-Term School of Physics and Astronomy Semester Two 2020/2021 Calculator instructions: You are allowed to use a calculator or a computer calculator in this assessment. Dictionary instructions: You are allowed to use your own dictionary in this assessment and/or the spell- checker facility on your computer. Assessment information: • This assessment is made up of 5 pages and is worth 30% of the module mark. • You have 48 hours to complete this open book online assessment. • You are recommended to take a maximum of 1 hour within the time available to complete the assessment. • You must answer all of the questions in this assessment. • You should indicate the final answer to each question by underlining it. At the end of each answer you should cite any websites or textbooks other than the course materials and recommended text books that you have used specifically to answer that question. You should always answer in your own words and not repeat material verbatim and you should explain each step of your working. • You must upload your answers via Minerva to GradeScope within the time allowed. You are advised to allow up to four hours to photograph your answers, and upload as a PDF to GradeScope. • When submitting your work, you must identify which questions are answered on which uploaded pages. You must also check that you have uploaded all the work you wish to be marked as part of this assessment and that the answers uploaded are clearly legible. Failure to do so may result in your work not being marked. • If there is anything that needs clarification or you have any problems, please email the module leader or
[email protected] and we will respond to you as quickly as possible within normal working hours UK time (9:00-17:00 hours, Monday-Friday). • This is a formal University assessment. You must not share or discuss any aspect of this assessment, your answers or the module more generally with anyone whether a student or not during the period the assessment is open, with the exception of the module leader and Physics exams team. Page 1 of 5 Turn the page over Module Code: PHYS237001 Approximate values of some constants Speed of light in a vacuum, c 2.998 × 108 m s−1 Electron Charge, e 1.602 × 10−19 C Electron rest mass, me 9.11 × 10−31 kg = 0.511 MeV c−2 Proton rest mass, mp 1.673 × 10−27 kg = 938.3 MeVc−2 Unified atomic mass unit, u 1.661 × 10−27 kg = 931.494 MeVc−2 Fine structure constant, α 1/137.036 Planck constant, h 6.626 × 10−34 J s Boltzmann constant, kB 1.381 × 10−23 J K−1 = 8.617 × 10−5 eV K−1 Coulomb constant, k = 1/4π�0 8.987 × 109 N m2 C−2 Rydberg constant, R 1.09373 × 107 m−1 Avogadro constant, NA 6.022 × 1023 mol−1 Gas constant, R 8.314 J K−1 mol−1 Stefan Boltzmann constant, σ 5.670 × 10−8 W m−2 K−4 Bohr magneton, µB 9.274 × 10−24 J T−1 Gravitational constant, G 6.673 × 10−11 m3 kg−1 s−2 Acceleration due to gravity, g 9.806 m s−2 Permeability of free space, µ0 4π × 10−7 H m−1 Permittivity of free space, �0 8.854 × 10−12 F m−1 1 Parsec, pc 3.086 × 1016 m Solar mass, M� 1.99 × 1030 kg Magnetic flux quantum, Φ0 2.0679 × 10−15 Wb Some SI prefixes Multiple Prefix Symbol Multiple Prefix Symbol 10−18 atto a 10−9 nano n 10−15 femto f 109 giga G 10−12 pico p 1012 tera T Page 2 of 5 Turn the page over Module Code: PHYS237001 Formula sheet for Vector Calculus Cylindrical coordinates x = ρ cosφ ρ = √ x2 + y2 y = ρ sinφ φ = tan−1(y/x) z = z z = z î = cosφ ρ̂− sinφ φ̂ ĵ = sinφ ρ̂+ cosφ φ̂ k̂ = ẑ ρ̂ = cosφ î+ sinφ ĵ φ̂ = − sinφ î+ cosφ ĵ r = ρ ρ̂+ z ẑ dr = dρ ρ̂+ ρdφ φ̂+ dz ẑ ∇f = ρ̂∂f ∂ρ + φ̂ 1 ρ ∂f ∂φ + ẑ ∂f ∂z ∇·V = 1 ρ ∂ ∂ρ (ρVρ) + 1 ρ ∂Vφ ∂φ + ∂Vz ∂z ∇× V = 1 ρ ∣∣∣∣∣∣∣∣ ρ̂ ρφ̂ ẑ ∂ ∂ρ ∂ ∂φ ∂ ∂z Vρ ρVφ Vz ∣∣∣∣∣∣∣∣ ∇2f = 1 ρ ∂ ∂ρ ( ρ ∂f ∂ρ ) + 1 ρ2 ∂2f ∂φ2 + ∂2f ∂z2 dS = ρ dρ dφ ẑ (flat surface) dS = ρ dφ dz ρ̂ (curved surface) dV = ρ dρ dφ dz Page 3 of 5 Turn the page over Module Code: PHYS237001 Spherical polar coordinates x = r sin θ cosφ r = √ x2 + y2 + z2 y = r sin θ sinφ φ = tan−1(y/x) z = r cos θ θ = tan−1 (√ x2 + y2 z ) r̂ = sin θ cosφ î+ sin θ sinφ ĵ + cos θ k̂ θ̂ = cos θ cosφ î+ cos θ sinφ ĵ − sin θ k̂ φ̂ = − sinφ î+ cosφ ĵ r = r r̂ dr = dr r̂ + rdθ θ̂ + r sin θdφ φ̂ ∇f = r̂∂f ∂r + θ̂ 1 r ∂f ∂θ + φ̂ 1 r sin θ ∂f ∂φ ∇·V = 1 r2 ∂ ∂r (r2Vr) + 1 r sin θ ∂(sin θVθ) ∂θ + 1 r sin θ ∂Vφ ∂φ ∇× V = 1 r2 sin θ ∣∣∣∣∣∣∣∣ r̂ rθ̂ r sin θφ̂ ∂ ∂r ∂ ∂θ ∂ ∂φ Vr rVθ r sin θVφ ∣∣∣∣∣∣∣∣ ∇2f = 1 r2 ∂ ∂r ( r2 ∂f ∂r ) + 1 r2 sin θ ∂ ∂θ ( sin θ ∂f ∂θ ) + 1 r2 sin2 θ ∂2f ∂φ2 dS = r2 sin θ dθ dφ r̂ (spherical surface) dV = r2 sin θ dr dθ dφ Vector derivative identities ∇(φψ) = φ∇ψ + ψ∇φ ∇·(φA) = A·∇φ+ φ∇ ·A ∇× (φA) = (∇φ) ×A+ φ(∇×A) ∇ · (A×B) = B·(∇×A) −A·(∇×B) ∇× (A×B) = A(∇·B) − (∇·A)B + (B·∇)A− (A·∇)B ∇× (∇×A) = ∇(∇·A) −∇2A. Page 4 of 5 Turn the page over Module Code: PHYS237001 SECTION A • You must answer all the questions from this section. • This section is worth 40 marks. • You are advised to spend 60 minutes on this section. 1. The density of charge of a material at position r = (x, y, z) is given by ρ(r) = sin(r2/a2), where r = √ x2 + y2 + z2 and a is real constant. Calculate the gradient of density ρ(r). Determine a vector at position r = (0, 0, 1) that points towards zero slope of charge density. [8 Marks] 2. Define the total surface Sc of a cylinder of radius R and hight H in terms of cylindrical coordinates. Express the surface element dS on the cylinder in cylindrical coordinates and explain its form using words and diagrams. Determine the area of the cylinder by evaluating the integral ∫ Sc dS. Calculate also ∫ Sc dS and explain why these two surface integrals are not equal. [8 Marks] 3. Evaluate the surface integral ∫ π 0 dφ ∫ φ 0 dθ cos θ. What is the surface of integration? Rewrite the integral with the order of θ and φ integrations interchanged and explain the procedure you follow. Show that interchanging the order of integration gives the same result. [8 Marks] 4. Show that∇×V = 0 for V = 3x2î + (2z − 7y4)̂j + (2y + 12z)k̂. Determine the scalar φ(r) such that V =∇φ. [8 Marks] 5. Demonstrate Green’s theorem,∫ S (∇×V) · dS = ∮ C V · dr, for the upper hemisphere of a sphere S of radius R when V(r) = yî− xĵ. [8 Marks] Total 40 marks Page 5 of 5 End Module Code: PHYS237001 Module Title: Maths 3 c© UNIVERSITY OF LEEDS End of Module Assessment School of Physics and Astronomy Semester Two 2020/2021 Calculator instructions: You are allowed to use a calculator or a computer calculator in this assessment. Dictionary instructions: You are allowed to use your own dictionary in this assessment and/or the spell- checker facility on your computer. Assessment information: • This assessment is made up of 8 pages and is worth 70% of the module mark. • You have 48 hours to complete this open book online assessment. • You are recommended to take a maximum of 1.5 hours within the time available to complete the assessment. • You must answer all of the questions in this assessment. • You should indicate the final answer to each question by underlining it. At the end of each answer you should cite any websites or textbooks other than the course materials and recommended text books that you have used specifically to answer that question. You should always answer in your own words and not repeat material verbatim and you should explain each step of your working. • You must upload your answers via Minerva to GradeScope within the time allowed. You are advised to allow up to four hours to photograph your answers, and upload as a PDF to GradeScope. • When submitting your work, you must identify which questions are answered on which uploaded pages. You must also check that you have uploaded all the work you wish to be marked as part of this assessment and that the answers uploaded are clearly legible. Failure to do so may result in your work not being marked. • If there is anything that needs clarification or you have any problems, please email the module leader or
[email protected] and we will respond to you as quickly as possible within normal working hours UK time (9:00-17:00 hours, Monday-Friday). • This is a formal University assessment. You must not share or discuss any aspect of this assessment, your answers or the module more generally with anyone whether a student or not during the period the assessment is open, with the exception of the module leader and Physics exams team. Page 1 of 8 Turn the page over Module Code: PHYS237001 Approximate values of some constants Speed of light in a vacuum, c 2.998× 108 m s−1 Electron Charge, e 1.602× 10−19 C Electron rest mass, me 9.11× 10−31 kg = 0.511 MeV c−2 Proton rest mass, mp 1.673× 10−27 kg = 938.3 MeV c−2 Unified atomic mass unit, u 1.661× 10−27 kg = 931.494 MeV c−2 Fine structure constant, α 1/137.036 Planck constant, h 6.626× 10−34 J s Boltzmann constant, kB 1.381× 10−23 J K−1 = 8.617× 10−5 eV K−1 Coulomb constant, k = 1/4π�0 8.987× 109 N m2 C−2 Rydberg constant, R 1.09373× 107 m−1 Avogadro constant, NA 6.022× 1023 mol−1 Gas constant, R 8.314 J K−1 mol−1 Stefan Boltzmann constant, σ 5.670× 10−8 W m−2 K−4 Bohr magneton, µB 9.274× 10−24 J T−1 Gravitational constant, G 6.673× 10−11 m3 kg−1 s−2 Acceleration due to gravity, g 9.806 m s−2 Permeability of free