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Field Studies
You may have noticed that the electric force, like gravity, acts at a distance.
It is Maxwell’s theory of electromagnetic phenomena which will resolve the
problem of this spooky action at a distance. But, like Maxwell, we must first
become familiar with the model Michael Faraday conceived.
As a first pass, it will be useful to look at a gravitational case, which is more
tangible. Below is a topological map of Mt. Hood. The contours are lines of
equal altitude.
Recall that the
force of gravity
depends on the
square of the
distance
between the
interacting
objects. One
of the objects
is the Earth.
Keeping this in
mind…..
[1] If you were to find curves in the Mt. Hood area where the force of
gravity was the same, what would they be?
[2] How do these curves compare to curves of equal potential energy?
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[3] If a ball were dropped at the “n” on the Newton Clark Gulch below the
HOOD on the map, what direction would it roll? How does that direction
relate to the contours?
[4] Now let’s simplify Mount Hood to a smooth cone. Below, draw the
contours for the cone.
On your sketch above, use dashed lines to show the directions balls would
roll if released from various places near the top.
You have just produced a map of the gravitational field on the surface of
the cone. Notice how the contours intersect the path lines at right angles.
This will be true of all the fields we investigate. The contours are examples
of equipotentials, curves of equal potential energy. The path lines become
lines of force, to which the force on a test object is tangent. These two
curves intersect at right angles.
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Suppose two equipotentials crossed. Would that be a problem? How would
you interpret the crossing?
Suppose two lines of force crossed. Would that be a problem? What would
that say about the direction of the force at the point of intersection?
Now, for that old friend, superposition. Suppose we have a single positive
electric charge. What would the equipotentials for the electric field around
it be?
What would the lines of force look like?
Now suppose we bring a second positive charge near the first, both firmly
anchored and unmoving. How would you go about determining the
equipotentials for the combination? Try your method another page and then
draw in the lines of force, making sure they intersect the equipotentials at
right angles.
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