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Unformatted text preview: 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 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 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 2 π integraldisplay 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 Q 2 π 2 π integraldisplay dφ ′ | vector r- vector r ′ | (4) Since vector r and vector r ′ are not necessarily in the same direction, we cannot simply...
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