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Coulomb Law Fields

Module by: George Brown. E-mail the author

Summary: This is designed as a challenging PLTL workshop in calculus-based introductory physics. It may also serve as as module in workshop physics. This is intended to indicate to the student the level of mathematical skill appropriate upon finishing the introductory courses in physics, and to encourage them to reach for this level of mathematical rigor.

Coulomb Law Fields - Page 1 of 6.

Introduction

Electric Field of a Single Point Charge

Consider the Coulomb force on a small test charge qq due to a point charge QQ: F E q = k Q q r 2 r ^ F E q = k Q q r 2 r ^ , where r r is the displacement vector from the location of QQ to that of qq , and kk is the Coulomb constant. The electric field in the space around QQ is defined in a formal way by the limit E Q [ r ] = Limit q 0 [ F q q ] = k Q r 2 r ^ E Q [ r ] = Limit q 0 [ F q q ] = k Q r 2 r ^ . (In this document, we use the convention that the arguments of functions are contained in square brackets [ ].)

An alternative form replaces the Coulomb constant with the more fundamental constant ϵ 0 ϵ 0 , called the permittivity of free space, and related to the Coulomb constant by k = 1 4 π ϵ 0 k = 1 4 π ϵ 0 . The numerical value is ϵ 0 ≈ 8.854× 10 - 12 C 2 N · m 2 ϵ 0 ≈ 8.854× 10 - 12 C 2 N · m 2 . This replacement gives us the most basic form of the Coulomb electric field of a single point charge QQ, as a function of the displacement r r from QQ:

Equation 1

E Q [ r ] = Q 4 π ϵ 0 r 2 r ^ E Q [ r ] = Q 4 π ϵ 0 r 2 r ^
(1)

Use of the basic Coulomb electric field, Equation 1, has an important advantage over the use of the basic Coulomb force, F E q = k Q q r 2 r ^ F E q = k Q q r 2 r ^ . Equation 1 represents a condition of space everywhere in the neighborhood of the point charge QQ. The electric field exists at all points in space surrounding the electric charge. If a test charge qq is placed at any location r r , an electric force F E q [ r ] = q E [ r ] F E q [ r ] = q E [ r ] acts upon it.

Since the work of Albert Einstein (1879 - 1955) in 1905, we (scientists) are convinced that the electric field of a point charge is real, and that electric charges actually change the nature of the space in which they are embedded. Equation 1, rather than simply representing a formal mathematical trick, makes an important statement about the reality in which we live. A very similar statement holds for the gravitational field that fills space in the neighborhood of mass. (We know almost intuitively that a ball, say, suspended in the air and released, will fall due to the gravitational force of the earth. We don't have to actually place a ball there and release it to know that a gravitational field exists at that point, and will provide a force on a test mass placed there.)

In using this Equation 1, it is important to have clearly in mind the meaning of the symbols that appear in it, especially that of the unit vector r ^ r ^ . In the basic relationships of electricity and magnetism, the unit vectors that appear always point from the location of source points to the location of field points. Field points are locations at which your field calculations apply. Source points locate the electric charges (or currents) used in your calculation. Equation 1 describes the electric field E [ r ] E [ r ] at the field point r r relative to the source point at the charge QQ. Thus the unit vector r ^ r ^ always points away from QQ. In the basic equations, distances (such as rr in Equation 1) always mean distances from source points to field points.

Taken literally, Equation 1 has very limited applicability - it only describes the electric field around a single, isolated point charge. Equation 1 cannot be used as it stands for any other situation. However, we can apply ideas from vector algebra and calculus to Equation 1 to derive Coulomb expressions for the electric field created by any stationary distribution of electric charge in three-dimensional (3D) space. (The electric field produced by a distribution of stationary charges is called the electrostatic field.)

Figure 1 presents a graphical representation of the electric field in the neighborhood of a single isolated point charge. In this figure, an orange sphere represents the central point charge, and the green arrows represent the electric field vectors at 26 equally spaced field points on the surfaces of three spheres of equally spaced radii. The lengths of the arrows in Figure 1 are proportional to the magnitude of the electric field vectors, while the arrow lengths in Figure 2 are proportional to the negative logarthim of the magnitude of E [ r ] E [ r ] .

Note that Figure 1 is accurate, in the sense that the lengths of the arrows reflect the actual magnitude of the electric field, but it may not be very informative, at least not visually. This is because the electric field magnitude goes like r - 2 r - 2 ; it decreases quite fast as the distance from the charge increases. The arrows on the sphere closest to the charge in Figure 1 are scaled so that they don't overlap or hide arrows further away. Then the arrows of the second sphere outward are twice as far away, and so are only one-fourth as long. They look like only the tiniest of arrows. And the arrows on the outer sphere are three times as far away as those on the inner sphere; they are only one-ninth as long, and they just look much like dots.

By contrast, Figure 2 gives a better visual idea of how the electric field gets weaker as one gets further away from the charge, but does so only by artificially lengthening the arrows that lie farther away from the charge.

Figure 1. An accurate portrayal of selected electric field vectors in the neighborhood of a single point charge.

Figure 2. As for Figure 1, except that more distant vectors are enlarged by a logarithmic factor for visual clarity.

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