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)=14πϵN∑n=1qn|r−rn|
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)=14πϵN∑n=1ρl(rn)|r−rn|Δl
Taking the limit as Δl→0 yields:
V(r)=14πϵ∫Cρl(l)|r−r′|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)=14πϵN∑n=1ρs(rn)|r−rn| Δs
Taking the limit as Δs→0 yields:
V(r)=14πϵ∫Sρs(r′)|r−r′| 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)=14πϵN∑n=1ρv(rn)|r−rn| Δv
Taking the limit as Δv→0 yields:
V(r)=14πϵ∫Vρv(r′)|r−r′| dv
where r′ represents the varying position over V with integration.