midterm4 - Activity 2 Solution for electric potential due...

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Activity 2: Solution for electric potential due to a ring Find the electrostatic potential in all space due to a ring with total charge Q and radius R V ( vector r ) = 1 4 πǫ 0 N summationdisplay i =1 q i | vector r - vector r i | (1) For a ring of charge this becomes V ( vector r ) = integraldisplay ring 1 4 πǫ 0 λ ( vector r ) | dvector r | | vector r - vector r | (2) where vector r denotes the position in space at which the potential is measured and vector r denotes the position of the charge. In cylindrical coordinates, | dvector r | = R dφ , where R is the radius of the ring. Thus, V ( vector r ) = 1 4 πǫ 0 2 π integraldisplay 0 λ ( vector r ) R dφ | vector r - vector r | (3) Assuming constant linear charge density for a ring with charge Q and radius R, λ ( vector r ) = Q 2 πR Thus, V ( vector r ) = 1 4 πǫ 0 Q 2 π 2 π integraldisplay 0 | vector r - vector r | (4) Since vector r and vector r are not necessarily in the same direction, we cannot simply leave | vector r - vector r | in curvilinear coordinates and integrate directly. One solution
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