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Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Striking Gold

Age 16 to 18
Challenge Level


This problem involves a retrospective look at Geiger and Marsden's famous experiment concerning the deflection of alpha particles passing through gold film. In this problem, assume that alpha particles and gold nuclei are modelled by hard spheres and that the alpha particles deflect if and only if they strike a gold nucleus. Note that the problem is about making good order of magnitude approximations, rather than performing a numerical calculation with a specific 'correct' answer.

[For this problem note that $1$ femtometre, $1\textrm{ fm}$, $= 1000$ picometres = $1\times 10^{-15}\textrm{ m}$, gold assumes a face-centred cubic crystal structure, and that the radius of a gold atom is $135\textrm{ pm}$.]

Scattering experiments have been used to determine that the radius $r_A$ of a nucleus of an atom of atomic number $A$ can be approximated by
$$r_A = 1.2\times A^{\frac{1}{3}} \textrm{ fm}\;.$$
Think about why this expression might make sense. What does it say about the radius of an alpha particle? What about the radius of the nucleus of a gold atom?

If you shoot an alpha particle into a block of gold, what is the chance of it being deflected by the first gold atom it enters?

Geiger and Marsden fired high energy alpha particles at a thin sheet of gold and noted that around $1$ in $8000$ incident alpha particles were deflected to some degree.

How thick do you think that their sheet of gold was?


Towards the end of the 19th century scientists were beginning to understand that atoms of matter contained both positively and negatively charged components. It was suggested by J. J. Thompson in 1898 that atoms could be described as spheres of positive electricity in which small negatively charged electrons were embedded -- like a currant bun or 'plum pudding', as it was described at the time. In 1909 Geiger and Marsden tested this prediction by firing alpha particles at gold foil. As expected, most of the particles passed directly through the gold foil. However, they were very surprised to find that a small fraction of these particles were dramatically deflected -- Lord Rutherford described this as 'about as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you!'. The subsequent realisation that atoms contained a dense nucleus and a lot of empty space was an important step in the development of the understanding of atomic structure.

Why not read more about the development of the theory of Rutherford Scattering ?