Question

The vector \( \vec{A} = (2x + 1)\hat{i} + (x^2 - 6y)\hat{j} + (xy^2 + 3z)\hat{k} \) is a

• A. sink field
• B. solenoidal field
• C. source field
• D. None of these

Hint:

Divergence \(>0 \) implies source, \(<0 \) implies sink, \( = 0 \) implies solenoidal field.

Answer 1

The Correct Option is C

Step 1: Write down the components of the vector field \( \vec{A} \). \[ A_x = 2x + 1, \quad A_y = x^2 - 6y, \quad A_z = xy^2 + 3z \] Step 2: Calculate the divergence of the vector field \( \vec{A} \). \[ \nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \] \[ \frac{\partial A_x}{\partial x} = 2, \quad \frac{\partial A_y}{\partial y} = -6, \quad \frac{\partial A_z}{\partial z} = 3 \] \[ \nabla \cdot \vec{A} = 2 + (-6) + 3 = -1 \] Step 3: Interpret the result of the divergence.
Since \( \nabla \cdot \vec{A} = -1<0 \), the vector field has sinks.