Electrostatic Potential Energy

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The force on a charged particle in a static electric field is given by the definition of the electric field, E, as

Since it can be shown that the electric field can be expressed as the negative gradient of a scalar potential function F then it follows from Eq. (1) that

The work done on a charged particle as it moves over a curve at constant speed from point A to B as

The negative sign on the right hand side of the first equality expresses the fact that force is applied in the direction opposite to the force due to the field. The last equality on the right side of Eq.(3) follows from the work energy theorem. V is the potential energy of a charged particle in an electric field. Thus we find that V = qF is the relationship between electric potential and potential energy. The potential F for a point charge is given by

Taking the negative gradient of both sides of Eq. (4) we obtain

Consider now a system of three charges. Start with a single charge q₁ in isolation. Then bring in another charge q₂ from infinity to a position near q₁. Then finally bring in a third charge q₃ in from infinity to a position near both q₂ and q₁ as shown in Fig. 1 below

Then the total work done in assembling this system of charges is then equal to the total potential energy V of the system and found to be

Extending this to a arbitrary number of charges yields

Extending to a continuous system of charges gives

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