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Physics LibreTexts

5.13: Electric Potential Field due to a Continuous Distribution of Charge

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The electrostatic potential field at r associated with N charged particles is

V(r)=14πϵNn=1qn|rrn|

where qn and rn are the charge and position of the nth 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 V(r) 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.

Continuous Distribution of Charge Along a Curve

Consider a continuous distribution of charge along a curve C. The curve can be divided into short segments of length Δl. Then, the charge associated with the nth segment, located at rn, is

qn=ρl(rn) Δl

where ρl is the line charge density (units of C/m) at rn. Substituting this expression into Equation ???, we obtain

V(r)=14πϵNn=1ρl(rn)|rrn|Δl

Taking the limit as Δl0 yields:

V(r)=14πϵCρl(l)|rr|dl

where r represents the varying position along C with integration along the length l.

Continuous Distribution of Charge Over a Surface

Consider a continuous distribution of charge over a surface S. The surface can be divided into small patches having area Δs. Then, the charge associated with the nth patch, located at rn, is

qn=ρs(rn) Δs

where ρs is surface charge density (units of C/m2) at rn. Substituting this expression into Equation ???, we obtain

V(r)=14πϵNn=1ρs(rn)|rrn| Δs

Taking the limit as Δs0 yields:

V(r)=14πϵSρs(r)|rr| ds

where r represents the varying position over S with integration.

Continuous Distribution of Charge in a Volume

Consider a continuous distribution of charge within a volume V. The volume can be divided into small cells (volume elements) having area Δv. Then, the charge associated with the nth cell, located at rn, is

qn=ρv(rn) Δv

where ρv is the volume charge density (units of C/m3) at rn. Substituting this expression into Equation ???, we obtain

V(r)=14πϵNn=1ρv(rn)|rrn| Δv

Taking the limit as Δv0 yields:

V(r)=14πϵVρv(r)|rr| dv

where r represents the varying position over V with integration.


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