A family of radial fields is

PHYSICS · Subjective · 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

A family of radial fields is F=kr^/rn\mathbf F = k\hat{\mathbf r}/r^n, with k>0k>0 a constant and nn any real number. The section met three members: n=−1n=-1 (F=kr\mathbf F = k\mathbf r), n=0n = 0 and n=2n = 2. (a) Find ∇⋅F\nabla\cdot\mathbf F for r>0r>0 in Cartesian components. (b) Find it again from the general-coordinates divergence in spherical coordinates, stating the scale factors you use. (c) For which nn is the divergence zero away from the origin, for which positive and for which negative? Check your formula against the three fields the section worked. (d) For n=3n = 3 the arrows still point away from the origin, yet the divergence is negative. Explain why, using a small cell whose side walls run along the arrows, cut off by two pieces of sphere.

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