Question 1
The direction of vector \(\vec{A}\) is radially outward from the origin, with \(|\vec{A}| = kr^n\) where \(r^2 = x^2 + y^2 + z^2\) and \(k\) is constant. The value of \(n\) for which \(\nabla \cdot \vec{A} = 0\) is:
Solution
We know that for a vector \(\vec{A}\) in spherical coordinates, if \(\vec{A}\) has only a radial component \(A_r\) (and no angular dependence), the divergence is given by:
Equation
\[\nabla \cdot \vec{A} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 A_r)\]
Given \(A_r = k r^n\), we substitute this into the equation:
Equation
\[\begin{aligned}
\nabla \cdot \vec{A} &= \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \cdot k r^n) \\
&= \frac{1}{r^2} \frac{\partial}{\partial r} (k r^{n+2}) \\
&= \frac{k}{r^2} (n+2) r^{n+1} \\
&= k (n+2) r^{n-1}
\end{aligned}\]
For the divergence to be zero (\(\nabla \cdot \vec{A} = 0\)) for all \(r\):
Equation
\[n + 2 = 0\]
Equation
\[\implies n = -2\]
Final Answer
Correct answer: A.