Inside a long straight wire heated evenly by a current

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Gradient, divergence and curl

Problem

Inside a long straight wire heated evenly by a current, the steady temperature a distance ρ\rho from the axis is T=T0−cρ2,T0=350 K,c=2.5×105 K m−2.T = T_0 - c\rho^2,\qquad T_0 = 350\ \mathrm K,\quad c = 2.5\times10^{5}\ \mathrm{K\,m^{-2}}. ((a)) Using the cylindrical Laplacian for a field that depends only on ρ\rho, find ∇2T\nabla^2T. ((b)) Check (a) by writing ρ2=x2+y2\rho^2 = x^2+y^2 and using the Cartesian Laplacian. ((c)) A classmate writes ∇2T=∂2T/∂ρ2\nabla^2T = \partial^2T/\partial\rho^2. What does that give, and what has been left out? ((d)) What does the sign of ∇2T\nabla^2T say about the temperature at a point compared with the points round it?

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