Using suitable diagrams, briefly describe how the molecular structure of a piezoelectric

crystal enables it to be employed to i) generate and ii) detect ultrasound pulses.                [4]

A  doctor working  in a hospital Accident  & Emergency  department  is responsible  for deciding whether the following patients should be sent to either the MRI unit, the x-ray CT unit, or the ultrasound department for further examination:

i)     A two-month-old baby who may have swallowed a wedding ring.

ii)    An elderly man who is suspected of have suffered a partial blockage of blood vessels in the brain.

iii)   A young woman who has splinters of glass embedded in her abdomen.

Giving reasons for your choice, state which diagnostic procedure is most appropriate in each

case, and why the other options may not be so helpful.                                                          [6]

Calculate the Doppler shift frequency obtained when a 3 MHz ultrasound probe is employed to investigate a vessel with a mean blood flow velocity of 50 cm/s, orientated at 60° to the

beam direction. The speed of sound in tissue is 1540 m/s.                                                     [2]

One sample ofModern Physics worth 40% (24 marks out of 60) of open-book exam which can be completed in 72 minutes (40% of 3 hours).

Question 3

Solutions

1 (a) [5 marks]

A descriptive account with a figure like the attached is required. Key points to make would include     (i) a brief account of pre-main sequence (MS) collapse (ii) burning CNO vs pp-chain with substantially

different MS lifetimes (iii) red giant, horizontal branch (He-burning) and asymptotic giant branch        phases but (iv) 1 ⊙  star does not ignite C/O/Si burning and ends as a cooling degenerate white dwarf while the 10 ⊙   star proceeds to a core-collapse supernova leaving a neutron star remnant.

(b)   [4 marks]

Demonstration of physical significance   of the main sequence turnoff which is     age-dependent because of the                   temperature-dependent efficiency of       nuclear fusion reactions (∝ % for pp-     chain in low mass stars and ∝  '(  for     CNO cycle in high mass stars). Massive  stars burn hydrogen much quicker so       leave the main sequence first. Since all    stars in a cluster can be assumed to be     co-eval, there is a clear main-sequence    turnoff which models can be used to age- date the turnoff point.

(c)  [7 marks]

Helium ignition occurs at the uppermost tip of the red giant branch after which the star descends onto a stable He-burning `main sequence’ called the horizontal branch. The energetics of the given reaction

are:

 = (2./0  −  2/3)  6  = (2 ´ 4.0026032 – 8.005305)6  = - 9.87´ 10-5  6  = - 91.9 keV

The surprising thing about this reaction is that it is endothermic, i.e. energy must be input to initiate it. A high core temperature (108 K) is thus required to generate the average kinetic energy of the gas        particles c.f. hydrogen fusion.

(d)   [4 marks]

The Schwarzschild radius of a black hole is 9:;  = 2= 6  so for the one at the Galactic centre 9:;  = (2 ´ 6.67 10- 11  ´ 2. 1030 ´ 4. 106)/ 9. 1016  = 1.18 1010 m = 17.0 solar radii

Average density =  = 4. 106  ´ 2 1030 kg / ( 4.189 ´ (1.18 1010)3) m3 = 1.16 106 kg m-3 (e)    [4 marks]

From Kepler’s law the period P of a star orbiting the black hole is related to the orbital radius r and the central mass M by

6  = 4F

so for a star whose period is 10 yrs, the orbital radius will be

r3 = (6.67 10- 11  ´ 4. 106  ´ 2. 1030  (10 ´ 365 ´ 24 ´ 3600)2 ´ 12.57) = 6.66 1044

So r = 8.74 1014 m (= 5825 A.U.)

At a distance of 8.5 kpc = 2.64 1020 m, this subtends an angle of only q = 3.31 10-6 radians which is

q ´ 180 ´ 3600/p = 0.68 arcsec. Tracing such an orbit requires exquisite angular resolution given           typical ground-based `seeing’ is 0.5- 1.0 arcsec. This work has been accomplished using adaptive optics which corrects for the blurring in the Earth’s atmosphere using a laser-generated artificial star.

2.       a)

The diagrams and the description must indicate the following:

•  A piezoelectric crystal contains a regular structure of molecules (electric dipoles) aligned     along a particular direction. Each molecule is physically elongated with a different electrical charge at each end. [ 1 mark]

•  When the crystal is placed in an electric field the dipoles attempt to align with the field by  rotating about their centres. This causes the crystal lattice to elongate in the direction of the field. [ 1 mark]

•   In generation mode, the sudden application of an electric field produces a sudden distortion of the crystal. This produces a vibration = an ultrasound pulse. [ 1 mark]

•   In detection mode, a mechanical compression (or elongation) of the crystal causes the dipoles to rotate. This changes the electrostatic charge on the opposite surfaces of the crystal. If the    surfaces are joined in a circuit, electrons in the connecting wire will flow. [ 1 mark]

b)       i) Ultrasound is the best option for detecting a ring swallowed by the infant. It is safe and   would show excellent contrast between the ring and surrounding soft tissues because of the large differences in acoustic impedance. [ 1 mark]

X-ray CT would be effective, but the dose of x-rays would be particularly hazardous for a baby. [ ½ mark]

MRI would not be appropriate for two reasons: the metallic ring would produce strong       image artefacts and the infant is unlikely to remain motionless during a scan lasting several minutes. [ ½ mark]

ii) Either X-ray CT or MRI would probably detect a partial block of blood vessels,   although both may require a contrast agent. X-ray CT is probably more likely as it is quicker and less expensive. [ 1½ marks]

Ultrasound would not be appropriate due to absorption/reflection by the skull. [½ mark] iii) All three options could be justified. Each has potential benefits and drawbacks. [ ½   mark]

X-ray CT would reveal the glass splinters with high resolution and contrast (necessary if the splinters are very small), but the x-ray dose would be high and not appropriate if the young woman is pregnant. [½ mark]

The MRI would not show small splinters, and splinters themselves would not produce an MRI signal. Nevertheless, MRI is safe and would give 3D images. [ ½ mark]

Ultrasound would be safe, and effective if the splinters are near the surface and are not too numerous (reflections by splinters near the surface would prevent ultrasound reaching       deeper splinters). [ ½ mark]

c)       Doppler shift frequencyfd = ‒2 u f cosθ / v                                                               [ 1 mark] = 2 x 0.50 x 3 x 106 x 0.5 / 1540 = 974 Hz                                                           [ 1 mark]

Solution to Question 3

(a)

In the photoelectric effect, the minimum energy that is needed to eject an electron from the surface of a metal is

F

.

 = ℎIJK  =  Φ  ⇒  IJK   =  =   = 1. 17  ×  10'[  

[2] points

(b)

[2] points for each of (i), (ii),  [1] point for (iii). (labelled below as a,b,c]

wavelengths  λ =  c/f

(i)  λ = 1.62 x10-27 m

(ii) λ = 1.73 x 10-35 m

(iii)λ = 4.78 x 10-26 m

(c)          (i) 2 points (1 for each case)

c)    (ii) 2 points (1 for each case)

For 1J:

For 1eV:                          = 1.6 10-16  s