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DEPARTMENT OF PHYSICS AND ASTRONOMY Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is available. Spring Semester 2014-2015 PHYSICS PHY009: 3 HOURS Answer questions ONE, FOUR and SEVEN plus ONE other from each section, SIX questions in all. Answers to different sections must be written in separate books, the books tied together and handed in as one. All questions are marked out of twenty. The breakdown on the right-hand side of the paper is meant as a guide to the marks that can be obtained from each part. Please clearly indicate the question numbers on which you would like to be examined on the front cover of your answer book. Cross through any work that you do not wish to be examined.

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SECTION A

1. COMPULSORY

(a) (i) Define simple harmonic motion (SHM). [2]

(ii) Sketch a graph showing the variation of acceleration with displacement for simple

harmonic motion.

[2]

(b) Monochromatic light with wavelength of 633 nm passes through a pair of narrow,

parallel slits 0.40 mm apart. Calculate the spacing of the bright fringes formed on a

screen 1.75 m away from the slits.

[3]

(c) A standing wave can be set up in a pipe, open at both ends, by sounding a tuning fork

of appropriate frequency at one of the open ends. The lowest frequency at which this

occurs for a pipe of length 25 cm is 686 Hz.

Calculate:

(i) The speed of sound in the pipe. [2]

(ii) The frequency of the second harmonic. [3]

(d) A convex lens with a focal length of 10 cm has an object placed at a distance of

18 cm from the optical centre.

(i) Calculate the distance of the image from the optical centre. [3]

(ii) Calculate the magnification of the above image. [2]

(iii) State the type of image formed. [1]

(e) Define the term dispersion as applied to light travelling through an optical medium. [2]

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2.

(a) A mass of 600 grammes is supported by a vertical spring. When the mass was

applied it extended the spring by 10 cm.

(i) Calculate the elastic constant of the spring.

[3]

The mass performs simple harmonic motion after it is displaced by an additional

4.0 cm and released.

Calculate:

(ii) The frequency of oscillation of the mass. [4]

(iii) The maximum velocity of the mass. [2]

(iv) The acceleration of the mass when the spring is at its equilibrium position. [1]

(b) A diffraction grating has 450 lines per millimetre. The second order diffraction angle

from a monochromatic light source incident upon this grating is 40˚.

Calculate:

(i) The spacing of the lines on the grating. [2]

(ii) The wavelength of the source. [4]

(iii) The value of the first order diffraction angle. [2]

(c) Define the angular magnification of an optical instrument. [2]

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3.

(a) Explain the meaning of the term resonance. [2]

(b) A stretched string of length 67 cm vibrates at its third harmonic with a frequency of

927 Hz.

(i) Illustrate the vibration by drawing a diagram showing the positions of the nodes and

antinodes.

[3]

The mode of vibration is now changed to the fundamental mode.

Calculate:

(ii) The wavelength of the fundamental mode. [2]

(iii) The frequency of the fundamental mode. [2]

(c) (i) A ray of light in air is incident at an angle of 30.0˚ on a block of ice. Calculate the

refractive index of the ice, given that the angle of refraction is 22.4˚. You may

assume that the refractive index of air is the same as that for a vacuum.

[3]

This same ray of light had travelled a distance of 0.5 m from its source before

entering the block of ice. The block of ice has a thickness of 0.6 m.

Calculate:

(ii) The distance that the light travelled within the block of ice before exiting back into

air.

[2]

(iii) Calculate how long it took for the light to travel from its source to the point where it

exited the block of ice.

[6]

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SECTION B

4. COMPULSORY

(a) Platinum, in the solid state, has a number density of 6.62 × 1028

atoms per cubic

metre. If its relative atomic mass is 195.1, obtain a value for the density of solid

platinum.

[4]

(b) Draw the unit cell for the fcc crystal structure. [2]

(c) Distinguish between elastic and plastic behaviour of a solid. [4]

(d) A bar of copper is 50 cm long at a temperature of 50 C. By how much does it expand

when heated to 150 C?

(The coefficient of thermal expansivity of copper = 1.7 × 105

K1

)

[4]

(e) A load of 40 N applied to a certain wire causes an extension of 0.2 mm. What is the

elastic energy stored in the wire?

[2]

(f) If the resistivity of silver is 15.9 nΩ m, calculate its conductivity. [2]

(g) State the majority carriers in p-type semiconductors. [2]

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5.

(a) A steel wire is 2.25 m long with a circular cross-section 0.5 mm diameter.

It is fixed vertically and has an extension of 4.0 mm when carrying a 7.0 kg load.

Calculate:

(i) The cross sectional area of the wire. [2]

(ii) The strain in the wire. [2]

(iii) The stress in the wire. [2]

(iv) The Young modulus of the wire. [2]

(b) (i) Sketch a graph showing the potential energy variation with separation (r) for a pair of

molecules. Indicate on your graph the equilibrium separation r0.

[4]

(ii) Define the dissociation energy ε and mark it on your graph to part 5(b)(i).

[2]

(iii) Sketch a second graph showing the variation of intermolecular force with separation

(r) for a pair of molecules. Indicate the equilibrium separation r0 on this graph.

[3]

(iv) A possible variation of force with separation is given by the equation

𝐹 =𝐴

𝑟13−

𝐵

𝑟7

Use this expression to obtain an equation for the equilibrium separation

r0 in terms of the constants A and B.

[3]

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6.

(a) Calculate the drift speed of electrons in a copper wire of diameter 1.0 mm2 when it is

carrying a current of 3.0 A. The number density of conduction electrons in copper is

1.0 1029

m3

.

[4]

(b) (i) Explain, with the aid of suitable diagrams, how the Hall Effect produces a small

transverse electric field when a magnetic field is applied to an electrical conductor,

and how the sign of the charge carriers affects the direction of the Hall Field.

[8]

(ii) If the Hall voltage is given by

𝑉𝐻 =𝐵𝐼

𝑛𝑒𝑡

Calculate the current which would have to be passed through a slice of n-type

germanium of thickness 0.50 mm with a transverse magnetic field of 0.8 T to obtain

a Hall voltage of 25 mV. Take the number density of electrons to be 6.0 1020

m3

.

[4]

(c) Explain the difference between intrinsic and extrinsic semiconductor materials. [2]

(d) Explain in terms of atomic behaviour, why the resistivity of aluminium increases

with increased temperature.

[2]

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SECTION C

7. COMPULSORY

(a) Define the electron-volt. [2]

(b) Electrons are accelerated in an X-ray tube through a potential difference of 75 kV.

Calculate the expected wavelength of the X-ray photons emitted by this tube.

[4]

(c) An oil drop of radius 1.6 μm carries a positive charge due to the loss of five

electrons. The density of the oil is 900 kg m3

. The drop is to be held stationary

between two horizontal parallel plates that are separated by a distance of 50 mm.

What voltage must be applied to the plates to keep the oil drop stationary?

[5]

(d) Calculate the wavelength of a neutron moving at a velocity of 8 105 ms

1. [2]

(e) Explain what is meant by the mass-defect of a nucleus. [2]

(f) State the nuclear constituents of the nuclide Pd46102 [1]

(g) State Einstein’s photoelectric equation, explaining the meaning of the symbols used. [4]

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8.

(a) Explain what is meant by the term nuclear fusion. [2]

(b)

(i)

(ii)

In an example of neutron induced fission of Uranium-235 the fission products are

isotopes of Barium and Krypton with an excess of neutrons.

n01 + U → Ba + Kr + 𝑋( n)0

1𝑍

8956

14492

235

Z and X are integer values.

Determine the values of Z and X

Calculate the energy released per fission, given the following masses.

U235 = 235.044 u

Ba144 = 143.923 u

Kr89 = 88.918 u

n01 = 1.001 u

[2]

[5]

(c)

A radioactive source of Cobalt-57 (57

Co) has an initial activity of 100 kBq and a half-

life of 272 days.

Calculate:

(i)

The decay constant of 57

Co, clearly stating its units.

[2]

(ii)

The time taken for the activity of the source to decrease to 25 kBq.

[2]

(iii) The original mass of the source [5]

(d) Name the two particles produced by beta minus decay [2]

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9.

(a) A metal surface emits photoelectrons at a constant rate when it is illuminated by a

certain monochromatic light source. Describe and explain how the emission of

photoelectrons will change when the frequency of the light source is increased

without changing its intensity.

[4]

(b) A Beryllium surface is illuminated by monochromatic light of wavelength 200 nm.

The work function of Beryllium is 5 eV.

Calculate:

(i) The work function of Beryllium in Joules.

[2]

(ii) The energy of the incoming photons.

[2]

(iii) The maximum kinetic energy of the emitted electrons.

[2]

(iv) The threshold wavelength of Beryllium.

[2]

(c) Describe the Rutherford model of the atom and the evidence that supported it [3]

(d) The atomic spectra of Hydrogen can be described by the Rydberg equation

1

𝜆= 𝑅 (

1

𝑛12 −

1

𝑛22) 𝑛2 > 𝑛1

The Rydberg constant R equals 1.097 × 107

m1

The Balmer series of spectral line emissions for hydrogen is given by 𝑛1 = 2.

Calculate:

(i) The wavelength of the first transition of the Balmer series. [3]

(ii) The shortest wavelength emitted by the Balmer series. [2]

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END OF EXAMINATION PAPER

PHYSICAL CONSTANTS &MATHEMATICAL FORMULAE

Physical Constants

electron charge e = 1.60×10−19 Celectron mass me = 9.11×10−31 kg = 0.511MeV c−2proton mass mp = 1.673×10−27 kg = 938.3MeV c−2neutron mass mn = 1.675×10−27 kg = 939.6MeV c−2Planck’s constant h = 6.63×10−34 J sDirac’s constant (~ = h/2π) ~ = 1.05×10−34 J sBoltzmann’s constant kB = 1.38×10−23 J K−1 = 8.62×10−5 eVK−1speed of light in free space c = 299 792 458 ms−1 ≈ 3.00×108 ms−1permittivity of free space ε0 = 8.85×10−12 Fm−1permeability of free space µ0 = 4π×10−7 Hm−1Avogadro’s constant NA = 6.02×1023 mol−1gas constant R = 8.314 Jmol−1K−1ideal gas volume (STP) V0 = 22.4 l mol−1gravitational constant G = 6.67×10−11 Nm2 kg−2Rydberg constant R∞ = 1.10×107 m−1Rydberg energy of hydrogen RH = 13.6 eVBohr radius a0 = 0.529×10−10 mBohr magneton µB = 9.27×10−24 J T−1fine structure constant α ≈ 1/137Wien displacement law constant b = 2.898×10−3 mKStefan’s constant σ = 5.67×10−8 Wm−2K−4radiation density constant a = 7.55×10−16 Jm−3 K−4mass of the Sun M� = 1.99×1030 kgradius of the Sun R� = 6.96×108 mluminosity of the Sun L� = 3.85×1026 Wmass of the Earth M⊕ = 6.0×1024 kgradius of the Earth R⊕ = 6.4×106 m

Conversion Factors1 u (atomic mass unit) = 1.66×10−27 kg = 931.5MeV c−2 1 Å (angstrom) = 10−10 m1 astronomical unit = 1.50×1011 m 1 g (gravity) = 9.81 ms−21 eV = 1.60×10−19 J 1 parsec = 3.08×1016 m1 atmosphere = 1.01×105 Pa 1 year = 3.16×107 s

Polar Coordinates

x = r cos θ y = r sin θ dA = r dr dθ

∇2 =1

r

∂

∂r

(r∂

∂r

)+

1

r2∂2

∂θ2

Spherical Coordinates

x = r sin θ cosφ y = r sin θ sinφ z = r cos θ dV = r2 sin θ dr dθ dφ

∇2 =1

r2∂

∂r

(r2∂

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1

r2 sin2 θ

∂2

∂φ2

Calculusf(x) f ′(x) f(x) f ′(x)

xn nxn−1 tanx sec2 x

ex ex sin−1(xa

)1√

a2−x2

lnx = loge x1x

cos−1(xa

)− 1√

a2−x2

sinx cosx tan−1(xa

)a

a2+x2

cosx − sinx sinh−1(xa

)1√

x2+a2

coshx sinhx cosh−1(xa

)1√

x2−a2

sinhx coshx tanh−1(xa

)a

a2−x2

cosecx −cosecx cotx uv u′v + uv′

secx secx tanx u/v u′v−uv′v2

Definite Integrals∫ ∞0

xne−ax dx =n!

an+1(n ≥ 0 and a > 0)

∫ +∞

−∞e−ax

2 dx =

√π

a∫ +∞

−∞x2e−ax

2 dx =1

2

√π

a3

Integration by Parts:∫ b

a

u(x)dv(x)dx

dx = u(x)v(x)∣∣∣ba−∫ b

a

du(x)dx

v(x) dx

Series Expansions

Taylor series: f(x) = f(a) +(x− a)

1!f ′(a) +

(x− a)2

2!f ′′(a) +

(x− a)3

3!f ′′′(a) + · · ·

Binomial expansion: (x+ y)n =n∑k=0

(n

k

)xn−kyk and

(n

k

)=

n!

(n− k)!k!

(1 + x)n = 1 + nx+n(n− 1)

2!x2 + · · · (|x| < 1)

ex = 1+x+x2

2!+x3

3!+ · · · , sinx = x− x

3

3!+x5

5!−· · · and cosx = 1− x

2

2!+x4

4!−· · ·

ln(1 + x) = loge(1 + x) = x− x2

2+x3

3− · · · (|x| < 1)

Geometric series:n∑k=0

rk =1− rn+1

1− r

Stirling’s formula: logeN ! = N logeN −N or lnN ! = N lnN −N

Trigonometry

sin(a± b) = sin a cos b± cos a sin b

cos(a± b) = cos a cos b∓ sin a sin b

tan(a± b) = tan a± tan b

1∓ tan a tan b

sin 2a = 2 sin a cos a

cos 2a = cos2 a− sin2 a = 2 cos2 a− 1 = 1− 2 sin2 a

sin a+ sin b = 2 sin 12(a+ b) cos 1

2(a− b)

sin a− sin b = 2 cos 12(a+ b) sin 1

2(a− b)

cos a+ cos b = 2 cos 12(a+ b) cos 1

2(a− b)

cos a− cos b = −2 sin 12(a+ b) sin 1

2(a− b)

eiθ = cos θ + i sin θ

cos θ =1

2

(eiθ + e−iθ

)and sin θ =

1

2i(eiθ − e−iθ

)cosh θ =

1

2

(eθ + e−θ

)and sinh θ =

1

2

(eθ − e−θ

)Spherical geometry:

sin a

sinA=

sin b

sinB=

sin c

sinCand cos a = cos b cos c+sin b sin c cosA

Vector Calculus

A ·B = AxBx + AyBy + AzBz = AjBj

A×B = (AyBz − AzBy) i+ (AzBx − AxBz) j+ (AxBy − AyBx) k = εijkAjBk

A×(B×C) = (A ·C)B− (A ·B)C

A · (B×C) = B · (C×A) = C · (A×B)

gradφ = ∇φ = ∂jφ =∂φ

∂xi+

∂φ

∂yj+

∂φ

∂zk

divA = ∇ ·A = ∂jAj =∂Ax∂x

+∂Ay∂y

+∂Az∂z

curlA = ∇×A = εijk∂jAk =

(∂Az∂y− ∂Ay

∂z

)i+

(∂Ax∂z− ∂Az

∂x

)j+

(∂Ay∂x− ∂Ax

∂y

)k

∇ · ∇φ = ∇2φ =∂2φ

∂x2+∂2φ

∂y2+∂2φ

∂z2

∇×(∇φ) = 0 and ∇ · (∇×A) = 0

∇×(∇×A) = ∇(∇ ·A)−∇2A