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10.1.2: What Are Fields?

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  • The idea of a field is rooted in the concept that there is some physical quantity that has a value “everywhere." The value can either change from location to location or can stay the same. Both fields that vary in space and fields that are constant in (regions of) space are important. A field can vary in time as well as space, so any field that we discuss is a function of both position and time. While this is an easy thing to state, it is rather abstract, so let us become more familiar with this definition by looking at some examples of fields:

    Scalar Fields

    Temperature Field

    The weather map above is similar to one you might see on the news; it represents a field.  There is not one universal temperature; the if you want to reference a temperature, you need to define both a time and a position where that temperature can be found. A question like “what is the temperature in San Francisco now?” can be answered because we have specified both the place and the time.  In other words, the temperature \(T\) is a function of position \((x,y,z)\) (for example), and time \(t\).

    With that being said, let's note a few important things about the map above:

    • It shows temperatures at a particular time, so what is being shown is \(T(x, y, t = \text{today})\). The full field \(T(x, y, t)\) could be represented by an entire archive of all previous (and future!) temperature maps.
    • On this map, the temperature is represented as a color, with the scale above giving the corresponding values.
    • On some weather maps the temperature is only shown for selected locations. Even in the places where a temperature is not labeled, a temperature can be measured.  There is a defined \(T\) for every shown value of \(x,y)\)
    Topographical Field

    A topography map represents the height of the Earth's surface as a function of position. Because the Earth does not shift quickly, we can neglect that this map depends on time.  Note that this map doesn't list height directly on the map like the temperature field did; it displays height by drawing lines along paths of equal height.  The land along a single contour is at the same elevation.  Drawing a line in a scalar field such that every point on the line has the same value in the field is an idea that will be utilized more later.

    Displacement Fields

    In unit 8 we characterized material waves by looking at \(y(x, t)\), which represents the displacement of the wave from its equilibrium position. The displacement depends on both the pertinent location (\(x\)) and time (\(t\)).  We now see that the wavefunction \(y(x, t)\) is a scalar field that describes the displacement of a material wave for all positions and times.  We call this kind of field a displacement field.

    Vector Fields

     The previous examples were scalar fields because they describe scalar quantities (like temperature or displacement in one dimension).  Vector Fields are different from the examples discussed above.  These fields define a vector, that is both a magnitude and direction, for all positions and times.  Vector Fields are much more relevant to the topics discussed later in this section, so take a while to make sure you can distinguish between the two.

    Wind Fields

    A wind map shows the velocity of the wind at various locations at a fixed time.  Because the velocity is a vector, this map must display both the direction and magnitude of wind.  The color is used to represent areas of the map where the wind is a fixed magnitude.  The arrows show the magnitude and direction of wind at various locations.  Like before, although the arrows aren't drawn everywhere on this map, the field defines a magnitude and direction for every point in the map.

    The representation of a vector field presented above, where arrows display the vector field at various locations, is called a field map representation.  We will become familiar with other ways to represent vector fields, each of which with its own advantages and disadvantages.  The advantage of field maps is that they can be read by eye very quickly, but it omits information about the vector field's scale .

    Warning: definitions can be taken too far

    There may be examples of functions that fit into our definition of "a field" which are probably not very useful. For example, one could define an elephant field in the following way: given a specific position and a specific time, the elephant field describes how many elephants exist there. Certainly this "field" fits into our definition, but it is not a terribly useful thing to consider. We cannot use the "elephant field" to make any new predictions, it is not required by any experiments, and it does not simplify any discussions. It is important to recall that we only consider something a field where it is required by experimental evidence (like the electric and magnetic fields) or it is convenient for discussing the quantity (like temperature).