Definition of Divergence
Divergence is a mathematical operation performed on a vector field to determine how much the vectors are spreading out or converging at a particular point in space. It is denoted by the symbol "∇ · V", where ∇ is the del operator and V is the vector field. The divergence of a vector field is a scalar quantity.

Solenoidal Field
A vector field is said to be solenoidal if its divergence is zero everywhere in space. In other words, the vectors in a solenoidal field do not spread out or converge at any point. Instead, they circulate around closed loops, resulting in a net flow of zero.

Explanation of the Correct Answer
The correct answer to the given question is option 'B', which states that a vector field with a vanishing divergence is called a solenoidal field. This is because a solenoidal field is characterized by its divergence being identically zero throughout the entire space.

Significance of Solenoidal Fields
Solenoidal fields play an important role in various areas of physics, particularly in the study of fluid flow and electromagnetism. In fluid dynamics, solenoidal fields are associated with the conservation of mass and the absence of sources or sinks. In electromagnetism, solenoidal fields arise from the absence of electric charges or magnetic monopoles.

Alternative Options
Let's briefly discuss the other options provided in the question and why they are not the correct answer:

- Rotational field: A rotational field is a vector field in which the curl is non-zero. It describes the circulation or rotation of vectors around a point. This is not the correct answer because the question specifically asks for a field with a vanishing divergence, not curl.
- Hemispheroidal field: There is no standard definition or widely recognized concept called a "hemispheroidal field" in physics. Therefore, this option is not the correct answer.
- Irrotational field: An irrotational field is a vector field in which the curl is zero. It represents a field with no circulation or rotation. Although this is related to solenoidal fields, it is not the correct answer because the question specifically asks for a field with a vanishing divergence, not curl.