FIELDS

Lots of information is difficult to visualise. If you cannot visualise data, it is extremely hard to make valid conclusions about it. Humans are visual animals. A field picture is a method of turning data into something visual. When geologists or biologists make Field Surveys to measure such quantities as species density or rock types in some part of a forest or geographical region. The resulting measurements are then usually depicted in colours or graphical means - each colour being some range of values obtained.

Physicists take the idea a little further - or back depending on your view. Physical measurements throughout space can be made and depicted graphically in the same fashion as other scientific surveys. As physical measurements fall - for our purposes - into two classes, scalar and vector, two physical field types can be constructed - vector and scalar.

SCALAR FIELDS

One of the simplest scalar fields is from ordinary map work - CONTOUR LINES - where the height above sea level is the scalar value.

[ Actually, height is a vector but as we only work in one direction, up - it is close enough to a scalar.
Much more usefully, height x g x m = Gravitational Potential Energy of m , a contour map is a map of GRAVITATIONAL POTENTIAL ENERGY ! This is certainly a scalar.]

The contours are lines of equal height - equal potential energy - and it is these that give the picture which is so useful. When bushwalking - we should be able to translate the contour lines back into the terrain around us.

Weather maps use similar lines of equal values - this time PRESSURE at Earth's surface. Values are depicted as ISOBARS. If you listen to weather forecasters - they translate the isobars into geographical "highs", "lows" and "ridges" - and they visualise these maps as if the weather was a system of hill and valleys.

This is a very useful technique to carry through into other fields such as Electrostatic fields.

SCALAR FIELDS are identifiable by the lines or colour coded areas joining places of one value or range of measurements. For the contour maps - the contour lines each represent a height. For the above weather map, isobars join points of equal pressure. For the top map, regions of similar UV energy are joined.

VECTOR FIELDS - Gravity Field, Electrostatic Field, Magnetic Field, River Velocity Field, Air Flow in a wind tunnel.

In all of the above examples, the measured quantity is a vector - a Force in the case of the first three and a velocity for the last two.

At any point in the field, we have BOTH a sense of SIZE and DIRECTION. Every point can have an arrow ( eg a "force arrow") as a representation. This form of visualization is clumsy.

We often use "Field Lines" instead - smooth continuous lines which when close represent large values and when widely separated represent weak values.

The direction and size of "flow" is usually in terms of a suitable "test object" - for any of the "forces" it must be an object which can be "forced" by the force we are measuring.

Gravity - a "small" test mass

Electrostatics - a "small" positive charge

Magnetism - a "small" magnetic wire

FASCINATING AND USEFUL LINK

Whenever we set up a scalar field, we can derive a vector field from it and vice versa!

A contour map - we can think of rivers running at right angles to the contours - but rivers have speed and direction - they are vectors! What is more, they are always at right angles to contours! Even more startling - the speed of the river will depend on how close the contours are together!

In Electrostatics, these become equipotentials and Electrostatic Field Strength with the E field lines always being at right angles to the equipotentials.

Equipotentials are lines of equal "voltage" - equal electrostatic pot energy per unit charge.

When we look at the above diagram as a set of hill and valleys, for a positive test charge the scenery looks like the diagram below.

A positive "test charge" near the red positive charge will then be repelled and attracted from the Force view - but in the energy view, it will lose ELECTROSTATIC potential energy, qV and gain kinetic energy.

qV = 1/2 mv²

This is the basis of a charge accelerators such as in the back of a TV tube.
(Electrons are negative - what does this change in our diagrams below? )

The algebraic link between the two fields is based on good old SLOPE.

The slope of a hill = rise / run = h/x This gives a measure of the speed of our river.

For our electrostatic case

E = slope of pot energy/charge graph = ΔV/Δx

= Potential difference between green lines/ physical distance between the lines.

We find similar equations for gravity and other fields.