Question

The value of the constant \(a\) so that the vector \[ \vec{V}=(x+3y)\hat{i}+(y-2z)\hat{j}+(x+az)\hat{k} \] is solenoidal is

• A. \(-1\)
• B. \(1\)
• C. \(-2\)
• D. \(2\)

Hint:

A vector field is \[ \boxed{\text{Solenoidal} \iff \nabla\cdot\vec{V}=0.} \] Always compute \[ \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}. \]

Answer 1

The Correct Option is C

Concept: A vector field is said to be solenoidal if its divergence is zero. \[ \boxed{\nabla\cdot\vec{V}=0} \] where \[ \vec{V}=P\hat{i}+Q\hat{j}+R\hat{k}. \] Then \[ \nabla\cdot\vec{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}. \]

Step 1: Identify the components. \[ P=x+3y,\qquad Q=y-2z,\qquad R=x+az. \]

Step 2: Compute the divergence. \[ \frac{\partial P}{\partial x}=1, \] \[ \frac{\partial Q}{\partial y}=1, \] \[ \frac{\partial R}{\partial z}=a. \] Hence, \[ \nabla\cdot\vec{V} = 1+1+a = a+2. \]

Step 3: Apply the solenoidal condition. Since \[ \nabla\cdot\vec{V}=0, \] we get \[ a+2=0. \] Therefore, \[ \boxed{a=-2.} \] Hence, the correct option is \[ \boxed{(C).} \]