Question

The divergence of a 3-dimensional vector \(\frac{𝑟̂}{𝑟^3}\) (𝑟̂ is the unit radial vector) is:

• A. \(\text{Zero}\)
• B. \(-\frac{1}{r^4}\)
• C. \(\frac{1}{r^3}\)
• D. \(-\frac{3}{r^4}\)

Answer 1

The Correct Option is A

The vector field is \( \mathbf{F} = \frac{\mathbf{r}}{r^3} \), where \( \mathbf{r} = r\hat{r} \) and \( r = \sqrt{x^2 + y^2 + z^2} \). Using the divergence formula:

\[ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}, \] after computing partial derivatives and simplifying, we find:

\[ \nabla \cdot \mathbf{F} = 0. \]

Final Answer: The divergence of \( \frac{\hat{r}}{r^3} \) is Zero.