Vector Fields

Wherever you are on earth, gravity acts as a force pulling you and everything else down towards the center of the earth. Gravity doesn't act on just a few objects, but on anything and everything in the region surrounding a massive object, so the earth is said to have a gravitational field: you can draw an arrow representing the gravitational vector at any point in the field.

The earth doesn't expend any energy to create this field. All massive objects have associated gravitational fields, but most of them are so small that we don't notice them. Even your computer keyboard technically has gravity, but since it has about a septillionth (10 to the 24th power) the gravity of the earth, its gravity is insignificant.

But how can the earth make things move without using up any energy? If you lift an object up, the energy you use to lift it against gravity is stored as potential energy. When you let go, that potential energy is converted into kinetic energy, the energy of motion, and the object falls back down. Thus gravity is not a true force (a push or pull on an object, or mass x acceleration), but a potential difference between higher and lower locations that makes massive objects accelerate downwards. Weight is a force derived from an objects mass times acceleration due to gravity (9.81 meters per second per second at the earth's surface, often referred to as 1 g).

In order for an object to move upward (against gravity), something has to provide an upward force greater than the downward force of the object's weight. The blue arrow in the picture to the right represents weight, and the red arrow is the upward force that the rocket engine produces.

The difference of these two forces is a net upward acceleration shown by the green arrow. If the upward force were equal to the rocket's weight, the length of the green arrow would be 0 and so the rocket wouldn't move. If the upward force were less than the force of gravity, the green arrow would point downward, and the rocket would fall back down to the ground.

A magnetic field is a region of vectors produced by a magnet. The earth's core, for example, generates a magnetic field that reaches to the surrounding area of space. The earth's field has a distinctive shape, similar to that of bar magnets, shown by the field lines in the image above. Field lines are different from drawings of other vectors, since they curve to show how the vector changes through the entire field. The strength, or flux of the field is shown by how many field lines go through a region. Like a gravitational field, a magnetic field creates differences in potential that cause other objects to move. Instead of attracting objects to the center of the earth, however, a magnetic field makes smaller magnets in the field turn to align themselves with the field lines in their region. The stronger the field, the greater the tendency for magnetic particles to become aligned. Field lines on the surface of the earth run between the north and south magnetic poles, which are close enough to the North and South Poles to make compasses useful for telling direction.

A vector quantity called magnetic moment is a simpler way to indicate the strength and size of a magnetic field than drawing out field lines for every magnet. It is drawn as an arrow aligned with the overall direction of the field (for a bar magnet, it is aligned with the magnet itself), whose length indicates the strength of the magnet.

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