A field has physical (unit-vector) components ,

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

Conditions of Validity, Ch Gradient, divergence and curl without coordinates, and with any coordinates

Problem

A field has physical (unit-vector) components Fr^=0F_{\hat r}=0, Fθ^=Bsin⁡θrF_{\hat\theta}=\dfrac{B\sin\theta}{r}, Fϕ^=0F_{\hat\phi}=0 in spherical coordinates (r,θ,ϕ)(r,\theta,\phi), with B=4.0 T mB=4.0\ \mathrm{T\,m}. Someone computes its divergence at r=2.0 mr=2.0\ \mathrm{m}, θ=π/3\theta=\pi/3 by writing ∇⋅F=1r2sin⁡θ∂∂θ(sin⁡θ Fθ^)\nabla\cdot\mathbf F=\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\big(\sin\theta\,F_{\hat\theta}\big) and substituting Fθ^=Bsin⁡θ/rF_{\hat\theta}=B\sin\theta/r directly into this formula. ((a)) Find the error in this working and state which factor it is missing. ((b)) Give the correct value of ∇⋅F\nabla\cdot\mathbf F at this point, in T m−1\mathrm{T\,m^{-1}}.

Browse the problem library