Topic: Electromagnetics
For a point charge having electric flux density, $D=\frac{Q}{4\pi r^{2}}a_{r}$, where $a_{r}$ is the unit vector in radial direction; volume charge density $\rho_{v}$ is
- A. 1
- B. 0
- C. Infinite
- D. $\frac{Q}{4\pi}$
Answer: B. 0
Explanation: Using the divergence theorem in spherical coordinates:
$\nabla\cdot D = \frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}D_{r}) + \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta D_{\theta}) + \frac{1}{r\sin\theta}\frac{\partial D_{\phi}}{\partial\phi}$
$\nabla\cdot D = \frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{Q}{4\pi r^{2}}\right) = 0$ (everywhere except at $r=0$). Since volume charge density $\rho_{v} = \nabla\cdot D$, $\rho_{v}$ evaluates to 0.