In this page we present examples of gravitional and electric fields in two dimensions. (We restrict ourselves to two dimensions so that we can include pretty pictures.)
A Gravitational Field
Suppose that there is a dense point-like object at the origin in the xy plane. What is the force on a particle at point (x,y) in the xy plane? Assume that the masses of the object and particle are 1 and that the gravitional constant is 1, the force at (x,y) is given by the vector field:
The vector field is shown in the plot below.
This is an example of a conservative vector field. Let
Then,
An Electric Field
Consider the vector field:
Modulo constants, F(x,y) is the force on a charged particle due to two oppositely charged, infinite straight wires perpendicular to the xy plane at (1,0) and (-1,0). The charge density on the wires is constant.
This is another example of a conservative vector field. It can be shown that F(x,y)=grad f(x,y), where
Copyright © 1996 Department of Mathematics, Oregon State University
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