Given that \bar{D}=r^{2} \bar{a}_{r}+2 \operatorname{Sin} \theta \bar{a}_{\theta} in spherical coordinate system, where D is the electric flux density, find the charge density p?
Question 1.5.1: Given that D¯=r^2 a¯r+2Sin θ a¯θ in spherical coordinate sys...
\text { Given, } \vec{D}=r^{2} a_{r}+2 \sin \theta a_{\theta} and \vec{D}=D_{r} a^{-1} r+D \vec{a}_{\theta}+D_{\phi} \vec{a}_{\phi}
So,
\begin{aligned}&D r=r^{2} \\&D \vartheta=2 \sin \vartheta \\&D \varnothing=0\end{aligned}and,
\begin{array}{r}P_{V}=\nabla \cdot \vec{D}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} D r\right)+\frac{1}{r \sin \theta} \cdot \frac{\partial}{\partial \theta} \\\left(D_{\theta} \sin \theta\right)+\frac{1}{r \sin \theta} \frac{\partial D_{\phi}}{\partial \phi}\end{array}\begin{aligned}&=\frac{1}{r^{2}} 4 r^{3}+\frac{2}{r \sin \theta} 2 \sin \theta \cdot \cos \theta \\&=4 r+\frac{4}{r} \cos \theta \\&=4\left[r+\frac{1}{r} \cos \theta\right]\end{aligned}
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