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12 September 2003

Gravity and Electricity

by George Hrabovsky, President of MAST

News from MAST

We are now beginning to develop task lists for the science project ideas listed on our web site.  We only have a couple now, but we intend to continue to grow our list of project ideas and to develop new project task lists.

The Electric Potential

Recall from a previous column, "The Energy of a World," that we established a relationship for potential energy based on a force,

U = -∫F r .

In the case of Coulomb's law,

F = (k q_1 q_2)/r^3Overscript[r,^] .

this becomes the line integral

U = -∫F · r .

      = -∫ (k q_1 q_2)/r^3Overscript[r,^] · r .

Evaluating this integral, based on the results from, "The Energy of a World," we have

U = k q_1 q_2 (1/r) .

Theory Challenge from Last Week

Since we have established that between any two masses or charges there exists, respectively, a gravitational force and a Coulomb force, it seems reasonable that if we have more than two masses or charges that the forces would be established between all pairs of masses or charges. To find the force due to the system, we would have to add up the forces between all pairs.  For gravitation with i number of masses, we would have,

F_grav = G Underoverscript[∑, q = 1, arg3] Underoverscript[∑, r = 1, arg3] (m_qm_r)/r_ (q r)^2 .

Likewise for charges

F_Coulomb = k Underoverscript[∑, m = 1, arg3] Underoverscript[∑, n = 1, arg3] (q_mq_n)/r_ (m n)^2 .

Theory Challenge

How do we express the potential energy due to multiple masses or charges?

Created by Mathematica  (September 12, 2003)