Chapter 2: Charge Density Distributions & Field Distribution Inside & Out

Illustration 1

Charge distributions in nature always have a finite volume. The concepts of point charge or line charge are useful approximations for the case when the distance between the source charge and the observation point is far greater than the size of the source itself. However, it is sometimes useful to know what the field behavior is not only far outside the source but also inside it.

Figure 1 shows the cross-section of a charged cylinder of radius  mm and the surrounding space (vacuum). The cylinder has a uniform charge density . Outside the cylinder, the field is equivalent to that of an infinite line charge (which would be located at the axis of the cylinder) of line charge density  C/m. It behaves as ,  being the radial distance from the cylinder’s center, . This is well observed in Figure 2.

Notice, however, that inside the cylinder the E-field magnitude grows linearly. Why that is so can be easily shown using Gauss Law discussed in Chapter 3. Here, we only point out that our plot is not just an approximation: theory shows that the field strength is indeed a linear function of the radial distance inside the source.

It is interesting to note that this is true for spherical charges as well. In a uniformly charged sphere, the radial electric field has a strength which grows as , while outside the sphere it behaves exactly like the field of a point charge:  (see Figure 3).

 
Figures (click to enlarge)
Figure 1

Figure 2

Figure 3

 

 

Figure 1

Figure 2

Figure 3