Question

If $\overline{r} = xi + yj + zk$ and $r = |\overline{r}| \neq 0$ then div. ($r^{n}\overline{r}$) =}

• A. $(n+3)r^{n}$
• B. $(n+1)r^{n}$
• C. $3r^{n}$
• D. $nr^{n}$

Hint:

$\text{div}(r^n \overline{r})$ is a standard identity often appearing in electromagnetic and gravitational problems.

Answer 1

The Correct Option is A

Step 1: Concept
Use the identity $\nabla \cdot (\phi \overline{A}) = \phi(\nabla \cdot \overline{A}) + (\nabla \phi) \cdot \overline{A}$.

Step 2: Meaning
Let $\phi = r^{n}$ and $\overline{A} = \overline{r}$. We know $\nabla \cdot \overline{r} = 3$ and $\nabla(r^{n}) = nr^{n-2}\overline{r}$.

Step 3: Analysis
$\nabla \cdot (r^{n}\overline{r}) = r^{n}(3) + (nr^{n-2}\overline{r}) \cdot \overline{r} = 3r^{n} + nr^{n-2}(r^{2})$.

Step 4: Conclusion
Combining terms: $3r^{n} + nr^{n} = (n + 3)r^{n}$. Final Answer: (A)