Solenoidal and irrotational fields

Question

I was wondering if there is any possibility that a vector field is irrotational and solenoidal at the same time. Can you help me out? I think I can't see the correct interpretation. I think that if, given a vector field A, it is solenoidal, it can't be irrotational. Could you give me some arguments?

Answer 1

Suppose a vector field $\vec A$ is irrotational,

$\nabla \times \vec A = 0, \tag 1$

and also solenoidal,

$\nabla \cdot \vec A = 0; \tag 2$

(1) implies the existence of a function $\phi$ such that

$\vec A = \nabla \phi; \tag 3$

then by (2),

$\nabla^2 \phi = \nabla \cdot \nabla \phi = \nabla \cdot \vec A = 0; \tag 4$

$\phi$ is thus a harmonic function; we thus see that a field satisfying (1) and (2) is the gradient of some harmonic $\phi$; conversely, if (3) binds for some harmonic $\phi$, then

$\nabla \cdot \vec A = \nabla \cdot \nabla \phi = \nabla^2 \phi = 0, \tag{5}$

and

$\nabla \times \vec A = \nabla \times \nabla \phi = 0, \tag{6}$

identically.

We thus see that the class of irrotational, solenoidal vector fields conicides, locally at least, with the class of gradients of harmonic functions.

Such fields are prevalent in electrostatics, in which the Maxwell equation

$\nabla \times \vec E = -\dfrac{\partial \vec B}{\partial t} \tag 7$

becomes

$\nabla \times \vec E = 0 \tag{8}$

in the event that the magnetic field $\vec B$ is constant in time; the in the absence of charges, e.g., in free space, the equation

$\nabla \cdot \vec E = \dfrac{\rho}{\epsilon_0} \tag 9$

yields

$\nabla \cdot \vec E = 0; \tag{10}$

it thus follows that there exists a (voltage) potential $\phi$ with

$\vec E = \nabla \phi \tag{11}$

and

$\nabla^2 \phi = 0. \tag{12}$

For more, see this wikipedia page on conservative vector fields, as well as this one on solenoidal fields.