Question

Given Vector Field:

\[ \vec{A} = (bx+4y^2z)\hat{i} + (x^3\sin z-3y)\hat{j} - (e^x+4\cos(x^2y))\hat{k} \]

For a vector field to be solenoidal,

\[ \nabla \cdot \vec{A}=0 \]
 

• A. \(2\)
• B. \(3\)
• C. \(1\)
• D. \(0\)

Hint:

For a solenoidal vector field, always put \(\nabla\cdot\vec{A}=0\).

Answer 1

The Correct Option is B

Concept:
A vector field is solenoidal if its divergence is zero. \[ \nabla\cdot\vec{A}=0 \]

Step 1: Write the vector field. \[ \vec{A}=(bx+4y^2z)\hat{i}+(x^3\sin z-3y)\hat{j}-(e^x+4\cos x^2y)\hat{k} \]

Step 2: Find divergence. \[ \nabla\cdot\vec{A} = \frac{\partial}{\partial x}(bx+4y^2z) + \frac{\partial}{\partial y}(x^3\sin z-3y) + \frac{\partial}{\partial z}[-(e^x+4\cos x^2y)] \] \[ \nabla\cdot\vec{A}=b-3+0 \] \[ \nabla\cdot\vec{A}=b-3 \]

Step 3: Apply solenoidal condition. \[ b-3=0 \] \[ b=3 \] \[ \therefore \text{Correct Answer is (B)} \]