Paper 3, Section I, B

Vector Calculus | Part IA, 2009

What does it mean for a vector field F\mathbf{F} to be irrotational ?

The field F\mathbf{F} is irrotational and x0\mathbf{x}_{0} is a given point. Write down a scalar potential V(x)V(\mathbf{x}) with F=V\mathbf{F}=-\nabla V and V(x0)=0V\left(\mathbf{x}_{0}\right)=0. Show that this potential is well defined.

For what value of mm is the field cosθcosϕreθ+msinϕreϕ\frac{\cos \theta \cos \phi}{r} \mathbf{e}_{\theta}+\frac{m \sin \phi}{r} \mathbf{e}_{\phi} irrotational, where (r,θ,ϕ)(r, \theta, \phi) are spherical polar coordinates? What is the corresponding potential V(x)V(\mathbf{x}) when x0\mathbf{x}_{0} is the point r=1,θ=0r=1, \theta=0 ?

[ In spherical polar coordinates ×F=1r2sinθerreθrsinθeϕ/r/θ/ϕFrrFθrsinθFϕ].\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta} \mid \begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right] .

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