Question

Find \( \nabla\phi \) if \( \phi = \log r \), where \( r = |\vec{r}| \):

• A. \( \dfrac{\vec{r}}{r} \)
• B. \( \dfrac{\vec{r}}{r^{2}} \)
• C. \( \dfrac{\vec{r}}{r^{3}} \)
• D. \( 0 \)

Hint:

Use \( \nabla f(r)=f'(r)\,\vec r/r \). With \( f=\log r \), \( f'(r)=1/r \), so the answer is \( \vec r/r^2 \).

Answer 1

The Correct Option is B

Step 1: Let \( \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \) and \( r=\sqrt{x^{2}+y^{2}+z^{2}} \). A key identity is \( \nabla r = \dfrac{\vec{r}}{r} \), the unit radial vector.

Step 2: For any function of \( r \) alone, the chain rule gives
\[ \nabla f(r) = f'(r)\,\nabla r = f'(r)\,\frac{\vec{r}}{r}. \]

Step 3: Here \( f(r)=\log r \), so \( f'(r)=\dfrac{1}{r} \).

Step 4: Substitute:
\[ \nabla(\log r) = \frac{1}{r}\cdot\frac{\vec{r}}{r} = \frac{\vec{r}}{r^{2}}. \]
\[ \boxed{\, \nabla(\log r) = \dfrac{\vec{r}}{r^{2}} \,} \]