# Electric Motors And Generators/Vectors and Fields

The use of vectors and vector fields greatly simplifies the analysis of many electromagnetic (and indeed other) systems. Due to their usefulness, these concepts will be used extensively in this book. For this reason it will be useful to begin with a general treatment of the subject.

### Vectors

A vector is a quantitative which has both magnitude, direction and sense. The magnitude represents the vector's size or physical quantity. The direction represents the vector's position with respect to a reference axis. The sense represents the vector's orientation and it is represented by its arrow head. This contrasts with the definition of a scalar which has only magnitude. Examples of scalar quantities include temperature, resistivity, voltage and mass. In comparison, examples of vector quantities would include velocity, force, acceleration and position.

The most familiar and intuitive use of vectors is in the two-dimensional (x, y coordinates) or three-dimensional (x, y and z coordinates) Cartesian coordinate system.

### Fields

The term field has a general meaning in mathematics and physics, but here we will be referring only to the special cases of scalar and vector fields. Generally, a field is a region in space where the quantity in question, exist and its influence is being felt. A scalar field is a region of space in which each point is associated with a scalar value. A classic example of a scalar field is a temperature field in a heated block of material.

If some heat source is applied to a cube of a conductive material, such as a metal, the temperature in the block will be highest where the heat source is applied, dropping off as we move away from the source in any direction. At every position inside the block a value could be assigned which is the temperature at this point. These temperature values make up the scalar temperature field in the block. It might be that it is possible to model accurately these values with some mathematical function, but the field itself is simply the variation in the scalar quantity in the space occupied by the block.

A vector field differs from a scalar field in that it has not only a magnitude at every position, but also direction. A good example of a vector field is the velocity of the fluid flow in a winding river of changing width. Clearly at every point in the river, the velocity of the fluid will have a magnitude (the speed) which will be lower where the river is wide, and higher where the river is narrow. However, the flow will also have a direction which changes as the water is forced around the river bends. If we noted the fluid speed and direction everywhere in the river, the result would be a vector field of the fluid flow.

In addition to varying in space, fields can also vary in time. In the first example, if we started with a cold block and then applied the heat source, mapping the temperature field at set time intervals, it would be seen that the values of the temperature at every point would change as the heat conducted throughout the block over time. The result therefore is a scalar field that varies in the three dimensions of space and one of time.