Conduction in concentric spheres applies to which geometry?

• A. Planar slabs
• B. Cylindrical tubes
• C. Concentric spheres
• D. Finite rectangular prisms

Answer: (C)

The essential idea is that the math of heat conduction changes with geometry because the area through which heat flows and the variation of temperature with distance depend on the shape. When heat moves through concentric spherical shells, the problem is purely radial, with symmetry around a center point. For steady, isotropic material with no internal heating, Fourier’s law in spherical coordinates reduces to (1/r^2) d/dr ( r^2 dT/dr ) = 0. Solving this shows the temperature depends on radius as T(r) = A + B/r, and the heat transfer rate through a spherical shell is Q = -k 4π r^2 dT/dr, which remains constant with r. The r^2 factor in the area is what makes the spherical geometry unique, and is the reason this conduction form applies to concentric spheres. Other geometries wouldn’t give this same radial form: planar slabs have a constant cross-sectional area, cylindrical tubes involve different radial terms, and rectangular prisms require Cartesian treatment.