- Consider A System Where The Concentration Of Particles Is Given By The Time Dependent Scalar Field C X Y Z T Co 1 X 1 (46.29 KiB) Viewed 49 times
Consider a system where the concentration of particles is given by the time-dependent scalar field c(x,y,z,t) = co (1+x") + sin(ny) e-ot, where (c, y, z) are Cartesian coordinates, t is time, co and a are positive constants, and n > 0 is a modelling parameter. (a) Find the number of particles N(t) inside the unit cube with sides defined by the planes 1 = 0, x = 1, y = 0, y = 1, 2=0,2 = 1. (b) Calculate the vector field F = Vc. (c) Calculate the surface integral Ds = /F Φ. F.dA, where S is the surface of the cube defined in part (a). (Hint: Take the normal vector to each surface to be pointing outwards from the cube.) (d) Verify that Gauss's theorem is satisfied by the field F for the cube with boundary S. (e) If J=-Dc, is there any choice of the constant D and the parameter n for which the continuity equation is satisfied? n