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# Electric Dipoles

Consider a charge located at position vector , and a charge located at position vector . In the limit that , but remains finite, this combination of charges constitutes an electric dipole, of dipole moment (195)

located at position vector . We have seen that the electric field generated at point by an electric charge located at point is (196)

Hence, the electric field generated at point by an electric dipole of moment located at point is (197)

However, in the limit that , (198)

Thus, the electric field due to the dipole becomes (199)

It follows from Equation (154) that the scalar electric potential due to the dipole is (200)

(Here, is a gradient operator expressed in terms of the components of , but independent of the components of .) Finally, because electric fields are superposable, the electric potential due to a volume distribution of electric dipoles is (201)

where is the electric polarization (i.e., the electric dipole moment per unit volume), and the integral is over all space.   Next: Charge Sheets and Dipole Up: Electrostatic Fields Previous: Electrostatic Energy
Richard Fitzpatrick 2014-06-27