Fields as far as the Eye Can See

Mark

Jan 20, 2024

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Understanding Fields Through Everyday Examples: Height Above Mean Sea Level

In our previous discussion, we delved into the unique assortment of properties that make up every object in the universe. In this article, we look at what happens as a single property changes over a region of space.

Boondock Technologies Logo with a variety of object properties arranged in a word cloud around it.

An assortment of object properties arranged in a word-cloud around the Boondock Technologies, LLC company logo.

Even objects that seem identical are different due to their unique positions in space.  No two objects can be in the same position at the same time.  

To further our understanding of the world, let's focus on how a single property of an object changes across a region, whether in two or three dimensions. A ready example of this is the concept of "Height Above Mean Sea Level (AMSL)."

AMSL: A Daily Encounter

Every time you take a walk around the neighborhood or a hike in the mountains, you're interacting with the property of ground elevation, or height above mean sea level (AMSL) - it's the height of the ground relative to a tideless and waveless ocean.  

Let's consider Washington State in the United States. If you were to view it from above, determining the exact elevation at any specific point, especially away from the shoreline, would be challenging or even impossible. To illustrate this, please look at the image below where I plotted the AMSL for various points across the state. Although there's a slight misalignment with the actual locations, it illustrates that the elevation varies across different locations.

A map of Washington State with several hundred AMSL heights indicated.  The elevations are slightly offset from the actual measurements likely due to a miscalculation by the author converting from one projection to another. 

But here's the catch: despite having numerous data points, it's hard to grasp the true nature of the terrain. The answers to questions like "Where's the highest point?" or "Which area is relatively flat and suitable for farming?" or "If I wanted to plan a 3-day hike, what's the easiest route?" aren't clear. The numbers alone don't paint the full picture. What we need to understand is how these numbers change across distances and in relation to the neighboring positions.

More Data Points: Clarity or Clutter?

Imagine we increase our measurement points infinitely.  So that we know the value at every location in space. Will that enhance our understanding? Probably not -- adding more and more data points would lead to a cluttered, overwhelming image, packed with information but lacking in clarity.  A map of Washington State covered by a tight mesh of tens of thousands of measurement points.

A map of Washington State covered by a tight mesh of tens of thousands of measurement points.

What is a Field?

A field is a 2d or 3d region of space where the value of an object property is known through measurement against a standard, or calculated through some mathematical means. 

Fields aren't defined by the quantity and location of data points.  They are defined by the object property they represent in a defined area or volume: the "gravitational field" that surrounds a planet, the "magnetic field", that surrounds an inductor, the "electric field" captured between the plates of a capacitor, etc.

Fields are continuous -- a value exists at every measurement point and everywhere in between.

The number of measurement points numbers themselves are overwhelming, so we often use a variety of visualization techniques to make sense not just of the numbers, but usually how those numbers change.

Visualizing Fields Through Contour Plots: Reading Between the Lines

One such visualization tool is a contour plot.  Instead of focusing on individual data points, contour plots connect points of identical elevation. This method transforms our perception of the data.

If you're new to contour plots, the squiggles and curves might seem a bit confusing.  Don't get overwhelmed -- there are two rules:

  • Curves connecting points of the same value are called isolines or isocurves.
  • The elevation change between any two isolines is constant.  (A line marks every 100 m in elevation change, or 200 m, or 75 m.  The value represented by each line does not matter as long as the value is constant over the entire graphic)

a contour map of Washington State

A contour map of Washington State above a satellite image of Washington state.

In this example, isocurves connect locations of constant elevation. Lines packed tightly into a given area indicate rapid elevation changes, while sparsely lined areas signify flat terrain. With this understanding, identifying peaks, valleys, and the easiest paths across the state becomes intuitive.

Fields: More Than Just Numbers

What we've explored with AMSL is a fundamental concept in understanding fields. A field in physics is a way to describe a property (like elevation) that varies in some region of space. Instead of overwhelming ourselves with endless data points, we use tools like contour plots to visualize how these properties change and interact in the space around us.

Height above mean sea level is just an example of one field.  Remember that there are thousands of object properties to choose from.  And you can plot any one of those over a 2 or 3-dimensional region of space.

3D Fields

Flat screens and flat maps are often used to represent a non-flat world.  That makes graphic design and data representation difficult, since the lines and arrows in the front of an image block whatever lies behind them. 

A globe covered with thousands of gravity vector arrows.

The acceleration due to gravity fluctuates slightly over the surface of the earth. It is higher at the poles (illustrated with red arrows pointed towards the center of the earth) and lower around the equator (illustrated with blue arrows.)

Often, it makes sense to show one or several cross-sections of a 3D field rather than the entire 3D field.

Exploring the Invisible: 3D Fields and Local Variations in Gravity

While our previous discussion on fields was anchored in the easily visualized concept of height above mean sea level, many three-dimensional fields are invisible to the naked eye. Gravity is one such field. It's not a constant force pulling us down to the Earth; it varies across the planet, growing stronger near the poles and weakening near the equator.  The fluctuations are imperceptible to humans but measurable with scientific instruments.

A close-up image of a globe covered with thousands of gravity vector arrows.

A close-up image of a globe covered with gravity vector arrows.

The Gravity of the Situation

Gravity is the result of the cumulative force of attraction between one object with mass and all the massive objects that surround it. The Earth is not a perfect sphere and its mass is not evenly distributed. Mountains, valleys, and even variations in geological structures below the surface contribute to slight variations in gravity's pull. These differences are minute, but with precise sensors and instruments, we can record the variations and then map them.

The images above show the measured acceleration near the surface of the Earth. The arrows represent the direction and magnitude of gravity's pull at various points around the globe. A longer red-colored arrow indicates a stronger pull, while a shorter blue-colored arrow indicates a weaker pull.  (Colors were added because the variation is so slight, it adjusts arrow length by less than a pixel)

Visualizing the Invisible

The first image might resemble a sea urchin with its spines sticking out - each arrow represents the gravitational pull at that point. But the number and placement of each arrow was chosen arbitrarily.  There is an acceleration of gravity at those spots and everywhere in between.  In the second image, we zoom in closer to the surface, where the interplay of these forces becomes more complex. The varying lengths and directions of the arrows make visible a landscape of invisible forces that mold the behavior of everything from satellites in low-earth orbit to the trajectory of a ball in flight.

Gravity Fields and Everyday Life

In everyday terms, you've likely never noticed gravitational variations.  The variation is subtle -- a fraction of a percent over what you might experience in a typical elevator.  But scientific instruments can accurately measure this number, and changes in values can indicate subterranean deposits of oil and minerals.

Connecting the Dots: From AMSL to Gravity Fields

Just as contour lines helped us understand elevation changes over a landscape, the arrows in these images help us visualize the invisible contours of gravity's pull. These gravity maps enable us to see beyond the 2D surface of our world and into the dynamic 3D space that surrounds us.

As we move forward, keep in mind these invisible fields are not just academic concepts but are active players in the workings of our daily world, including the technology of antennas that we will explore in upcoming articles.