1.5 Calculating Electric Fields of Charge Distributions
1.5 Calculating Electric Fields of Charge Distributions
LEARNING OBJECTIVES
- Explain what a continuous source charge distribution is and how it is related to the concept of quantization of charge
- Describe line charges, surface charges, and volume charges
- Calculate the field of a continuous source charge distribution of either sign
The charge distributions we have seen so far have been discrete: made up of individual point particles. This is in contrast with a continuous charge distribution, which has at least one nonzero dimension. If a charge distribution is continuous rather than discrete, we can generalize the definition of the electric field. We simply divide the charge into infinitesimal pieces and treat each piece as a point charge.
Note that because charge is quantized, there is no such thing as a “truly” continuous charge distribution. However, in most practical cases, the total charge creating the field involves such a huge number of discrete charges that we can safely ignore the discrete nature of the charge and consider it to be continuous. This is exactly the kind of approximation we make when we deal with a bucket of water as a continuous fluid, rather than a collection of H2OH2O molecules.
Our first step is to define a charge density for a charge distribution along a line, across a surface, or within a volume, as shown in Figure 1.5.1.
(Figure 1.5.1)
Definitions of charge density:
- charge per unit length (linear charge density); units are coulombs per metre ()
- charge per unit area (surface charge density); units are coulombs per square metre ()
- charge per unit volume (volume charge density); units are coulombs per cubic metre ()
Then, for a line charge, a surface charge, and a volume charge, the summation in Equation 1.4.2 becomes an integral and is replaced by , , or respectively:
The integrals are generalizations of the expression for the field of a point charge. They implicitly include and assume the principle of superposition. The “trick” to using them is almost always in coming up with correct expressions for , , or as the case may be, expressed in terms of , and also expressing the charge density function appropriately. It may be constant; it might be dependent on location.
Note carefully the meaning of in these equations: It is the distance from the charge element to the location of interest, (the point in space where you want to determine the field). However, don’t confuse this with the meaning of ; we are using it and the vector notation to write three integrals at once. That is, Equation 1.5.2 is actually
EXAMPLE 1.5.1
Electric Field of a Line Segment
Find the electric field a distance above the midpoint of a straight line segment of length that carries a uniform line charge density .
Strategy
Since this is a continuous charge distribution, we conceptually break the wire segment into differential pieces of length , each of which carries a differential amount of charge . Then, we calculate the differential field c