Question

If \(\mathbf{f} = f_1(y, z) \hat{i + f_2(z, x) \hat{j} + f_3(x, y) \hat{k}}\), then \( \mathbf{f} \) is

• A. Irrotational
• B. Solenoidal
• C. Both Irrotational \& Solenoidal
• D. Gradient

Hint:

A solenoidal vector field has zero divergence, meaning that there is no net flow out of any region of space.

Answer 1

The Correct Option is B

We are given the vector field \( \mathbf{f} = f_1(y, z) \hat{i} + f_2(z, x) \hat{j} + f_3(x, y) \hat{k} \), where the components depend on the respective variables as shown. To determine the nature of the field, we need to check if it is:

• Irrrotational: A vector field is irrotational if \( \nabla \times \mathbf{f} = 0 \).
• Solenoidal: A vector field is solenoidal if \( \nabla \cdot \mathbf{f} = 0 \).

Let's calculate the divergence \( \nabla \cdot \mathbf{f} \):

\[ \nabla \cdot \mathbf{f} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z} \]

Given that \( f_1 \), \( f_2 \), and \( f_3 \) depend on different variables, the partial derivatives will simplify to the following:

\[ \frac{\partial f_1}{\partial x} = 0, \quad \frac{\partial f_2}{\partial y} = 0, \quad \frac{\partial f_3}{\partial z} = 0 \]

Thus, the divergence is zero:

\[ \nabla \cdot \mathbf{f} = 0 \]

Since the divergence of \( \mathbf{f} \) is zero, \( \mathbf{f} \) is a solenoidal field.

The field \( \mathbf{f} \) is solenoidal, and thus the correct answer is option (B).