What is the magnetic flux density when the vector potential is a position vector?
Since the given potential is a position vector, the gradient will be 3 and H = -3. Thus the flux density B = μH = 4π x 10-7 x (-3) = -12π x 10-7 units. Explanation: The Laplacian of the magnetic vector potential is given by Del2(A) = -μ J, where μ is the permeability and J is the current density.
What do you mean by vector potential?
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field.
Is vector potential unique?
Although we cannot express the magnetic field as the gradient of a scalar potential function, we shall define a vector quantity A whose curl is equal to the magnetic field: B=curl A=∇×A. 1 does not define A uniquely. …
What is scalar potential of a vector?
The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that: The first of these conditions represents the fundamental theorem of the gradient and is true for any vector field that is a gradient of a differentiable single valued scalar field P.
How do you know if a vector field is path independent?
A conservative vector field (also called a path-independent vector field) is a vector field F whose line integral ∫CF⋅ds over any curve C depends only on the endpoints of C. The integral is independent of the path that C takes going from its starting point to its ending point.
Does a vector potential exist?
In physics courses vector potentials come up because magnetic fields are divergence free and so have vector potentials. However, they typically take a different approach to finding these. They use the fact that usually a magnetic field is the result of a stream of charged particles called a current.
Does a vector field have potential?
In general, if a vector field P(x, y) i + Q(x, y) j is the gradient of a function f(x, y), then −f(x, y) is called a potential function for the field. Vector fields that have potential functions are called conservative fields.
What is the potential function of a vector field?
Definition 1.1 A potential function for a vector field F = is a function ϕ such that F = Vϕ. A vector field F is conservative if it has a potential function. Definition 1.2 A region R in R2 or R3 is connected if every pair of points in R can be connected with a continuous curve that lies entirely within R.
What is the physical significance of vector potential?
The physical meaning of the electric scalar potential is usually considered to be potential energy per unit charge. The physical meaning of the magnetic vector potential is actually very similar: it’s the potential energy per unit element of current.
How do you determine if a vector field is a gradient field?
The scalar function that a vector field is the gradient of is called the potential function of the vector field. It can be found by integrating the equations that define this relationship. For example, if I have the field ⟨y,x⟩, then a potential function f must satisfy ∂f∂x=y,∂f∂y=x.
What is the gradient of vector field?
The gradient of a function is a vector field. It is obtained by applying the vector operator V to the scalar function f(x, y). Such a vector field is called a gradient (or conservative) vector field. = (1 + 0)i +(0+2y)j = i + 2yj .
How do you prove a 3d vector field is conservative?
If a three-dimensional vector field F(p,q,r) is conservative, then py = qx, pz = rx, and qz = ry. Since F is conservative, F = ∇f for some function f and p = fx, q = fy, and r = fz.
What is curl of a vector field?
The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum “circulation” at each point and to be oriented perpendicularly to this plane of circulation for each point.
How do you solve a vector curl?
The curl of a vector field A, denoted by curl A or ∇ x A, is a vector whose magnitude is the maximum net circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum!.
What is the curl of a constant vector?
If F is a constant vector field then. curl F = 0 .
What does it mean if divergence is zero?
zero divergence means that the amount going into a region equals the amount coming out. in other words, nothing is lost. so for example the divergence of the density of a fluid is (usually) zero because you can’t (unless there’s a “source” or “sink”) create (or destroy) mass.
Why is the gradient of B Zero?
A scalar field is single valued. That means that if you go round in a circle, or any loop, large or small, you end up at the same value that you started at. In a scalar field there can be no difference, so the curl of the gradient is zero.
What is the divergence of curl of vector?
The divergence and curl can now be defined in terms of this same odd vector ∇ by using the cross product and dot product. The divergence of a vector field F=⟨f,g,h⟩ is ∇⋅F=⟨∂∂x,∂∂y,∂∂z⟩⋅⟨f,g,h⟩=∂f∂x+∂g∂y+∂h∂z.
What is the significance of divergence of a vector?
The divergence of a vector field simply measures how much the flow is expanding at a given point. It does not indicate in which direction the expansion is occuring. Hence (in contrast to the curl of a vector field), the divergence is a scalar.
What is the physical meaning of divergence?
The physical significance of the divergence of a vector field is the rate at which “density” exits a given region of space. By measuring the net flux of content passing through a surface surrounding the region of space, it is therefore immediately possible to say how the density of the interior has changed.