Electric Potential Field due to a Continuous Distribution of Charge

The electrostatic potential field at

associated with

charged particles is

where

and

are the charge and position of the

particle. However, it is more common to have a continuous distribution of charge as opposed to a countable number of charged particles. We now consider how to compute

three types of these commonly-encountered distributions. Before beginning, it’s worth noting that the methods will be essentially the same, from a mathematical viewpoint, as those developed in Section 5.4; therefore, a review of that section may be helpful before attempting this section.

Consider a continuous distribution of charge along a curve

. The curve can be divided into short segments of length

. Then, the charge associated with the

segment, located at

, is

where

is the line charge density (units of C/m) at

. Substituting this expression into Equation 5.13.1, we obtain

Taking the limit as

yields:

where

represents the varying position along

with integration along the length

.

Consider a continuous distribution of charge over a surface

. The surface can be divided into small patches having area

. Then, the charge associated with the

patch, located at

, is

where

is surface charge density (units of C/m

) at

. Substituting this expression into Equation 5.13.1, we obtain

Taking the limit as

yields:

where

represents the varying position over

with integration.

Consider a continuous distribution of charge within a volume

. The volume can be divided into small cells (volume elements) having area

. Then, the charge associated with the

cell, located at

, is

where

is the volume charge density (units of C/m

) at

. Substituting this expression into Equation 5.13.1, we obtain

Taking the limit as

yields:

where

represents the varying position over

with integration.