A vector field F⃗ 𝐹→ is called solenoidal ifa.div(curl(F⃗ ))=0𝑑𝑖𝑣(𝑐𝑢𝑟𝑙(𝐹→))=0b.curl(F⃗ )=0𝑐𝑢𝑟𝑙(𝐹→)=0c.div(F⃗ )=0𝑑𝑖𝑣(𝐹→)=0d.None of these

To determine which option correctly describes a solenoidal vector field, let's analyze the given statements:

1. Understanding Solenoidal Vector Fields: A vector field $\mathbf{F}$ is called solenoidal if it satisfies the condition of having zero divergence. In mathematical terms, this is written as: $\text{div}(\mathbf{F}) = 0$ This indicates that the field has no sources or sinks within the field, resembling incompressibility in fluid dynamics.

2. Analyzing the Options:

   • a. $\text{div}(\text{curl}(\mathbf{F})) = 0$: This statement is true for any vector field, as the divergence of a curl is always zero. However, it does not define a solenoidal field specifically.
   • b. $\text{curl}(\mathbf{F}) = 0$: This represents a condition for conservative vector fields; it does not pertain to solenoidal fields.
   • c. $\text{div}(\mathbf{F}) = 0$: This is the defining characteristic of solenoidal vector fields.
   • d. None of these: Since option 'c' is correct, this option is incorrect.

Final Answer

The correct answer is c. $\text{div}(\mathbf{F}) = 0$. A vector field $\mathbf{F}$ is solenoidal when it has zero divergence.