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# CLF Multiple Point Sources

Module by: George Brown. E-mail the author

Summary: Part of a PLTL workshop on Coulomb Law forces, pertaining to the Coulomb force law in cases of multiple point sources. Intended as part of the study of electricity and magnetism in undergraduate physics courses.

## Electric Force on a Field Charge due to Fixed Source Charges

Suppose there are more than one point source charges providing forces on a field charge. A fairly simple example with three source charges (shown in green and indexed by subscripts) and one field charge (in red, designated q q), is diagramed below. We assume that the source charges are fixed in space, and the field charge q q is subject to forces from the source charges.

Note the coordinate system that has been chosen. All of the charges lie on the corners of a square, and the origin is chosen to collocate with the lower right source charge, and aligned with the square. Since we can have only one origin of coordinates, no more than one of the source points can lie at the origin, and the displacements from different source points to the field point differ.

The total force on the field charge q q is due to applications of the force described in Equation (1): F E = k q 1 q 2 r 2 r ^ F E = k q 1 q 2 r 2 r ^ , from each of the source charges. So the total force is the sum of these individual forces. (In physics, when quantities add linearly in this way, we say that the quantities superpose. To say that the total force is the sum of the individual forces, we say that the forces are subject to superposition.)

The displacements of the field charge from each source charge are shown as light blue arrows in the diagram below.

Using the principle of superposition for this example, and therefore applying Equation (1) three times and summing the results, we find that

### Equation (2)

F E q = k q ( q 1 r 1 2 r ^ 1 + q 2 r 2 2 r ^ 2 + q 3 r 3 2 r ^ 3 ) F E q = k q ( q 1 r 1 2 r ^ 1 + q 2 r 2 2 r ^ 2 + q 3 r 3 2 r ^ 3 )
(1)

To actually use this relationship, we need to determine the values of the r n r n and the r ^ n r ^ n , where n = { 1 , 2 , 3 } n = { 1 , 2 , 3 } . To see how this can be done, it is useful to add to the figure the vectors that locate each of the source charges.

This figure repays some study. First note that the simple symbol r r has been reserved to mean the location of the field point. (This is a conventional usage.) Other locations are designated with subscripts and primes. The vectors from source points to the field point (needed in the use of Equation (1)) carry subscripts and are shown in light blue. The vectors locating source points (shown in brown in the figure) carry subscripts and a prime sign to indicate they are source locations. By virtue of the geometry of the diagram and choice of coordinate system, r 1 = 0 r 1 = 0 and r 1 = r r 1 = r . For each source point r n r n , there is a triangle formed by vectors: r n + r n = r r n + r n = r . This fact (which is always true for any three points in space) is the key for doing calculations for the force on the field charge, because the vectors needed in the use of Equation (1) are given for each source charge by r n = r - r n r n = r - r n .

Assume that the side of the square in this example is a a. Then for the first source charge r 1 = r = a y ^ r 1 = r = a y ^ . For the second source charge r 2 = a ( - x ^ ) r 2 = a ( - x ^ ) and r 2 = a ( x ^ + y ^ ) r 2 = a ( x ^ + y ^ ) . For the third source charge r 3 = a ( ( - x ^ ) + y ^ ) r 3 = a ( ( - x ^ ) + y ^ ) and r 3 = a x ^ r 3 = a x ^ . Substituting these values into Equation (2) provides the solution for this example

F E q = k q 2 a 2 ( ( 2 q 3 + q 2 ) x ^ + ( 2 q 1 + q 2 ) y ^ ) F E q = k q 2 a 2 ( ( 2 q 3 + q 2 ) x ^ + ( 2 q 1 + q 2 ) y ^ )
(2)
.

We can generalize Equation (2) to give a basic result for the Coulomb force on a field point charge q q due to an arbitrary number N N of fixed source point charges q n q n in a compact form:

### Equation (3)

F E q [ r ] = k q n = 1 N q n r n 2 r ^ n F E q [ r ] = k q n = 1 N q n r n 2 r ^ n
(3)

Equation (3), properly applied, provides the solution for the force on any point field charge affected by the electric force from an arbitrary number of fixed source charges with known values and locations.

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