Show that all vectors of a central field lie on rays through 0 , and that the magnitude of the vector field at a point depends only on the distance from the point to the center of the field. It is also useful to look at central fields which are not defined at the point 0 .

Imagine standing in the midst of a swirling gust of wind, with each breeze indicating a direction and having strength — that's what a vector field loosely represents in mathematics and physics. A vector field assigns a vector (a quantity with both magnitude and direction) to every point in space. In the context of a central field, this means that each location is associated with a vector that points towards or away from a central point.

For example, think about the Earth's gravitational field: it's a type of central vector field where all vectors point toward the Earth's center. Similarly, the electric field around a charged particle can also be described as a central vector field. Each vector in these fields gives us information about the force that a particle would experience if it were placed at that point in space.

In order to understand where each vector in a field is pointing, we need some way to identify each position in space. This is where the position vector comes in. A position vector, denoted often as \(\boldsymbol{r}\), extends from a point of reference, which is usually the origin, to a point in space. It essentially acts as a locational address for every point in a field.

The concept might seem complex, but it's quite like using GPS coordinates to locate a destination. In the case of a central field, the 'destination' is any point within the field, and the 'GPS coordinates' are encapsulated by the position vector. The position vector's length and direction hold the key to understanding not only where a point is but also in defining the characteristics of the vector field at that point.

Now, you might be wondering how to measure the strength of the vectors we've been talking about. This is described by the magnitude of a vector field. The magnitude equates to the force, intensity, or strength at a given point in the field and is a scalar quantity, meaning it only has magnitude but no direction.

For a central field, like the ones created by gravity or electric charges, the magnitude of a vector field at any particular point is determined by a function, generically represented as \(f(r)\), where \(r\) is the distance from the central point, or the origin, to the point of interest. The further away you are from the source, the weaker the magnitude becomes, reflecting an inverse relationship between distance and the strength of the field.

The concept of radial distance is fundamental when discussing central fields. Radial distance, denoted as \(r\), is the straight-line distance from the origin or center of the field to any point within the field. You can think of it like the radius of a circle, but extending out in every direction in three-dimensional space.

Radial distance is crucial because it dictates how the vectors change throughout the field. In a central field, all vectors change uniformly as you move further away from the center — they don't depend on the direction you are traveling from the center, only on how far. Understanding radial distance helps in comprehending why the magnitude of a central field's vector depends solely on this variable and not on the position around the field.