For a vector E, which one of the following statements is NOT TRUE? A. If $$\nabla $$ · E = 0, E is called solenoidal B. If $$\nabla $$ × E = 0, E is called conservative C. If $$\nabla $$ × E = 0, E is called irrotational D. If $$\nabla $$ · E = 0, E is called irrotational

The correct answer is D.

A vector field $E$ is called solenoidal if $\nabla \cdot E = 0$. This means that the divergence of $E$ is zero, which implies that there are no sources or sinks of the field. In other words, the total amount of electric field lines entering a closed surface is equal to the total amount of electric field lines leaving the surface.

A vector

field $E$ is called irrotational if $\nabla \times E = 0$. This means that the curl of $E$ is zero, which implies that there are no vortices in the field. In other words, the electric field lines do not form any closed loops.

A vector field can be both solenoidal and irrotational, but it can also be neither. For example, the electric field of a point charge is both solenoidal and irrotational. However, the electric field of an infinite line charge is solenoidal but not irrotational.

The statement “If $\nabla \cdot E = 0$, $E$ is called irrotational” is not true because a vector field can be solenoidal without being irrotational. For example, the electric field of a point charge is solenoidal but not irrotational.