What is meant by solenoidal vector field?

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: A common way of expressing this property is to say that the field has no sources or sinks.

What is solenoidal field give example?

Solenoidal fields are characterized by their so-called vector potential, that is, a vector field A such that a=curlA. Examples of solenoidal fields are field of velocities of an incompressible liquid and the magnetic field within an infinite solenoid.

What is solenoidal vector field and Irrotational field?

A Solenoidal vector field is known as an incompressible vector field of which divergence is zero. Hence, a solenoidal vector field is called a divergence-free vector field. On the other hand, an Irrotational vector field implies that the value of Curl at any point of the vector field is zero.

How do you prove a vector is irrotational?

A vector field F in R3 is called irrotational if curlF = 0. This means, in the case of a fluid flow, that the flow is free from rotational motion, i.e, no whirlpool. Fact: If f be a C2 scalar field in R3.

Which of the following is not vector field?

Answer: Speed is not a vector quantity. It has only magnitude and no direction and hence it is a scalar quantity.

Which vector field is solenoidal?

The lines of flow diverge from a source and converge to a sink. If there is no gain or loss of fluid anywhere then div F = 0. Such a vector field is said to be solenoidal. A key point: F is a vector and the curl of F is a vector.

Which vector is Solenoidal?

An example of a solenoid field is the vector field V(x,y)=(y,−x). This vector field is ”swirly” in that when you plot a bunch of its vectors, it looks like a vortex. It is solenoid since divV=∂∂x(y)+∂∂y(−x)=0.

Which of the following is an example of vector field?

A gravitational field generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere’s center with the magnitude of the vectors reducing as radial distance from the body increases.

What do you mean by irrotational vector field?

An irrotational vector field is a vector field where curl is equal to zero everywhere. If the domain is simply connected (there are no discontinuities), the vector field will be conservative or equal to the gradient of a function (that is, it will have a scalar potential).

Which of the following is NOT example of vector?

Power is not an example of a vector quantity. Power is energy (or work) per unit time, time does not consider in power so it’s not a vector quantity. Physical quantities which have both magnitude and direction are called vector quantities.

How do you prove a vector is conservative?

As mentioned in the context of the gradient theorem, a vector field F is conservative if and only if it has a potential function f with F=∇f. Therefore, if you are given a potential function f or if you can find one, and that potential function is defined everywhere, then there is nothing more to do.

Can a vector field be expressed as a solenoidal field?

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as:

What kind of field is a solenoid coil?

A solenoid is a long spiral coil of wire, usually cylindrical, through which a current can be passed to produce a magnetic field. More abstractly, let a be a vector field (on R 3 ) with div

What is the origin of the word solenoidal?

(Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition .) Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained as if in a pipe, so with a fixed volume.

What is meant by solenoidal vector field? In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: A common way of expressing this property is to say that…