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VECTOR CALCULUS: Consider R to be a sphere centred at the origin with boundary ρ = 2 in spherical coordinates.

COURSEWORK IN VECTOR CALCULUS
Coursework instructions Answer all parts of the question below in no more than 5 A4 sides. Submit a scanned copy of your report via Learn by 12 noon on Thursday the 19th of March. Submit only your own work.


Question 1.

Consider R to be a sphere centred at the origin with boundary ρ = 2 in spherical coordinates. Within it there is a spherical compartment that is vacuum with boundary ρ = 1. You may assume that the vacuum has density zero. The density of the materials inside the sphere where 1 ≤ ρ ≤ 2 is equal to 1 for z ≥ 0 and 2 otherwise.

(a) (16) Compute the z coordinate of the centre of mass of R. (b) (4) An X-ray source is positioned at a known point with spherical coordinates (ρs,φs,θs) (away from the sphere) emitting a beam of intensity Iin aligned to a vector that’s normal to the surface of the sphere pointing toward its interior. If the beam is attenuated according to the density of the sphere d as


log Iin /Iout =Zc dds, where c is the trajectory (line) of the beam and Iout its intensity as it comes out of the sphere, find Iout if Iin = 1 assuming that the beam does not attenuate outside the sphere.

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