A parallel X-ray beam traverses a uniform circular disc of radius R and linear attenuation coefficient 1 unit⁻¹. A ray is offset by distance t from the centre, with |t|≤R. If incident and emergent intensities are Iᵢ and Iₒ, what is ln(Iᵢ/Iₒ)?
The path through the disc is the chord length 2√(R²−t²). Beer–Lambert attenuation is Iₒ=Iᵢexp(−μℓ). Taking the natural logarithm gives ln(Iᵢ/Iₒ)=μℓ. With μ=1 this is the chord length, A. Only the central ray has length two R.
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