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 answe
that question. You should always answer in your own words and not repeat material
ve
atim 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 XXXXXXXXXX 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 ove
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 = XXXXXXXXXXMeVc−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 XXXXXXXXXX × 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, Φ XXXXXXXXXX × 10−15 W
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 ove
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
= 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
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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
)
̂ = 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 r̂
dr = dr r̂ + rdθ θ̂ + r sin θdφ φ̂
∇f = r̂∂f
∂
+ θ̂
1
∂f
∂θ
+ φ̂
1
sin θ
∂f
∂φ
∇·V = 1
2
∂
∂
(r2Vr) +
1
sin θ
∂(sin θVθ)
∂θ
+
1
sin θ
∂Vφ
∂φ
∇× V = 1
2 sin θ
∣∣∣∣∣∣∣∣
̂ rθ̂ r sin θφ̂
∂
∂
∂
∂θ
∂
∂φ
Vr rVθ r sin θVφ
∣∣∣∣∣∣∣∣
∇2f = 1
2
∂
∂
(
2
∂f
∂
)
+
1
2 sin θ
∂
∂θ
(
sin θ
∂f
∂θ
)
+
1
2 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 ove
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 answe
that question. You should always answer in your own words and not repeat material
ve
atim 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 XXXXXXXXXX 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 ove
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 = XXXXXXXXXXMeV 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 XXXXXXXXXX× 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