Physics Exam Answers: Autumn Semester 2003-4

Level One

F31AB1: Biophysics

2) 2 diopters

4) (c) Zm = 7.3 × 106 kg m-2 s-1

5) (a) 0.31 m.

6) (b) vmax = 1.36 ms-1 (c) Fmax = 7.8 × 10-7 N.

7) (a) N(t) = N0 er t ln 2 where r is division rate. (b) No. of ATP molecules = 7.56 × 1016

(c) 9.9 × 108 glucose molecules. (d) 13.5 hours.

8) (a) K+out = 10.3 mM (b) 1.24 × 109 dissociated ions. (c) Factor of 426.

F31AM1: Mechanics I

1) τ = 13 i + 5 j + 6 k Nm ; L = -32 i - 22 j - 18 k kgm2s-1.

2) Time taken = 43.1 s.

3) (r, θ, φ) = (3, 48.2º, 26.6º).

4) g = 0.64 ms-2 ; Mass dumped = 421 kg.

5) (a) I = ½M(R12 + R22) ; (b) I = ¼M(R12 + R22).

6) Acceln. = 2.35 ms-1 ;

7) Small θmax when sin θ ≈ θ. This does not need to be assumed in part (c).

8) (a) Coriolis – zero; Centrifugal – out from local centre (position on axis at that latitude). (b) Both in same direction as for centrifugal in part (a). (c) Coriolis is to the West; Centrifugal as for (a) and (b).

For vertical plumb line, plane flies West at 201 ms-1.

F31ARM/ F31YMC: Modelling

1) (a) Equilibrium position when x = √2. (b) Potential tends to ∞ as x tends to zero or ∞ and has minimum at V(x) = 4 and x = √2. (c) Force is unaffected by change in C.

2) Small time means tL/R .

3) Energy has units of Joules or kgm2s-2. Dimensions are [ML2T-2]; I has dimensions [MT-3]; Possible relation for I is h-3 ET4 c-2.

4) (b) Fractional change in period = 1.89 × 104.

5) (a) Dimensions of α are[M-1L4T]. (b) Steady state value for height of water in lower bucket at long times is zero. (c) a = αgρ/S ; b = h1 + h2 . (d) h3 rises from zero at t = 0 and tends asymptotically to (h1 + h2)/10 at long times.

F31AV1: Environmental Physics I

3) Max. power out = 1.1 × 103 W. TH= 170 ºC.

4) Power = 23.6 kW.

5) Age of solar system = 8.5 × 109 years.

7) Estimate of average power available: 75MW.

8) Average number of collisions = 83. Mean free path = 0.9 μm. Mean free time = 41 ns.

F31AZ1: Introduction to Electromagnetism

1) I3 = 3A ; I4 = 2A ; I5 = 1A.

2) L = 56.6 mH.

3) is is max. when ω = 1/√(LC).

4) Force = 2.25 × 10-12 N.

5) Power is max. when r = R.

7) (a) B = μ0I/2πd ; (b) Flux linkage = Nr2μ0I/2d ; (c) M = Nr2μ0/2d.

8) (c) V at drop surface = 104 V.

Level Two

F32AA4: The Structure of Stars

2) Average density = 7.5 × 105ρ.

6) (b) (i) RS (ii) LSMS5 . (c)

F32AB2: Biomedical Physics II

2) (i) Vol. of O2 = 4  10-11 ml.

4) Reading would need to be reduced by a factor of 1.017 .

6) vart = 0.034 ms-1 ; vcap = 3  10-4 ms-1 ; vven = 3.4  10-3 ms-1 . (a) ratio (single) = 0.2

(b) ratio (system) = 4.5 . Reynold’s number increased by a factor of 2.

7) At 100 kV, ratio is 5:1 . At 4MV, ratio is 0.5:1 . Dose per field per fraction is 2.69 Gy.

8) Functional Residual Capacity = 2 litres.

F32AM4: Elements of Mathematical Physics

1) For square wave A0 = 1 ; An = ; Bn = 0 .

2) I = 1 .

3) u(x, y) = .

4) Line integral = 14.4 .

5) .A = 4r – 2/r sin cos  .

6) G = (3x2y , x3 , 6z) ; .G = 6xy + 6 .

7) F.T. of f(t-τ) is e-iωtF(ω) ; F(ω) = ; G(ω) =

8) Bn = .

F32AQ1: Quantum Physics I

1) ΔE = 3.3 MeV .

3) Momentum eigenvalues are ± ħk ; probability of observing either is ½

4) (a) J = ; (b) J = 0 .

5) Eigenvalue = -3 .

6) Energy eigenvalue = ½ ħω ; C = ; = 0 .

7) Shift in energy of (a) ground state = ; (b) excited state = 0.

8) Wave fn. at t ≠ 0 is ; ;

Probability of E2 = ½ .

F32AZ2:Electromagnetism 1

3) Force on loop is along the y direction.

5) Possible potential is V = -kxy + constant.

6) Era = ; Era = .

7) Relationships are: E0/B0 = c where c = ω/k .

Time average of S is given by .

Area of smallest detectable object = 0.13 m2 .

8) Bra = ; Bra = ; By = .

Level 3

F33AA6: Cosmology

1) Creation of H atoms needed to maintain steady state is ~ 5 × 10-19 atoms m-3 s-1.

4) Redshift ~ 4 × 1012 .

5) RE(r0) = .

6) (b) .

7) (d) A ≈ 1010 Ks½ ; (e) Tf ~ 3 × 1010 K.

F33AB5: Medical Imaging

2) Percentage increase = 2% .

4) (d) Signal-to-Noise Ratio ≈ 4 .

5) (b) Transverse Magnetisation = ; Longitudinal Magnetisation = .

(c) Mz = M0 (1 - ) ; Mx =

6) (b) Frequency detected = 3.952 MHz.

7) Ejection fraction = 53% .

F33AN3: Further Nuclear Physics

1) Mass of exchange particle ≈ 100 MeV.

2) Diff. x-section = ; Total x-section = πA2 .

3) Plane separation = 3.90 × 10-10 m.

5) (a) allowed; (b) forbidden; (c) forbidden; (d) allowed.

6) Excitation Energy = 14.56 MeV.

7) l = 0

8) Total s-wave scattering x-section = 4π/k2 ; upper limit = 2.61 barns.

F33AQ3: Quantum Physics III

1) ΔE = 59.9 cm-1 .

3) = ½ ; ψ1: n=1, l=0,=0 ; ψ2: n=2, l=1,=-1; ψ3: n=3, l=1,=0 since

sin φ corresponds to 50% lz = 1 and 50% lz = -1.

4) 1× (J = 1); 2× (J = 2); 1× (J = 3); three transitions allowed.

5) ΔE = AF so splittings are 8, 7, 6, 5, 4, 3 and 2 × 10-2 cm-1 .

6) Energies of π-polarised lines are 22,539.04 and 22,537.16 cm-1 .

7) Ionisation energy of electron in He+ ion is 54.4 eV; ≈ 30 eV ;

; S = 0 for ground state of helium.

8) Energy of ground state = ; ΔE0 = 0.15 V0 ; If perturbation applied over entire

well, then ΔE0 = 0.

F33AS2: Solid State Physics II

6) (a) EC: 8U ; EV : 4U ; Gap = 2E0 – 6U . (b) me* = ; mh* = .

7) For intrinsic material, n = p = 1.81  1012 m-3 . For doped material, n = 660 m-3 and

p ≈ 5  1021 m-3 ; = 0.19 eV.

8) 0 = 1.09 V.

F33AS5: Atoms and Molecules at Surfaces

1) ; ; Cu(110)(21)-O .

2) Oxygen exposure is 15.1 L.

3) Branching ratio = 2.

5) Ediff = 0.15 eV.

6) 17 Bragg diffraction spots.

F33AZ3: Electromagnetism II

1) = 66 m; Vel. = 415 ms-1 .

2) Distance between nodes is 1 cm.

4) B0 = 3.47 T .

6) (a) E = ; H = ; (b) E = ; H = ; (c) E = H = 0 .

Radial dependence and direction of S : (a) radially inwards with dependence r ; (b) along

wire with dependence 1/r2 ; (c) zero.

7) Electric field at aerial is 17mV m-1 in vertical direction.

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