Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 76 / 2 / c / Solution

Use the same physical normalization for both fields. Part (b) has

E_{s} = E_{i} ∣ f_{\infty}^{(1)} ∣^{2} / R^{2} + O (R^{- 3})

, so the common energy flux prefactor cancels. The differential cross-section for scalar wave scattering is

σ_{d}^{(1)} (r, r_{0}) = ∣ f_{\infty}^{(1)} (r, r_{0}) ∣^{2} = \frac{k ^{4}}{16 π ^{2}} \int_{V} (n^{2} - 1) e^{- i k_{s} \cdot r} d^{3} r^{2} .

The far-field pattern has dimensions of length in this normalization, so the differential cross-section for scalar wave scattering has dimensions of area per unit solid angle. Only the scattered field is used in this ratio; interference between the incident and scattered fields is a different contribution to the total field's energy flux.

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