Electrostatic Potential and Capacitance

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• Summary

The external electric field E and the corresponding external potential V may vary from point to point.

In a uniform electric field, the dipole experiences no net force; but experiences a torque τ given by

τ = p x E

which will tend to rotate it (unless p is parallel or antiparallel to E.

The amount of work done by thee xternal torque will be given by

Taking θ⁰ as 90^° and θ₁ as θ, potential energy,
U(θ) = -p.E = - pEcosθ

Work done in bringing a charge q from infinity to the point P in the external field is qV.

This work is stored in the form of potential energy of q.

If the point P has position vector r relative to some origin, we can write:
The potential energy of q at r in an external field

= qV(r)

where V(r) is the external potential at the point r.

Work done in bringing the charge q₁ from infinity to r₁ is q₁ V(r₁). Consider work done in bringing q₂ to r₂.In this step, work is done not only against the external field E but also against the field due to q₁.

Work done on q₂ against the external field
= q₂V (r₂)
Work done on q₂ against the field due to q₁

where r₁₂ is the distance between q₁ and q₂.

By superposition principle for fields, add up the work done on q₂ against the two fields. Work done in bringing q₂ to r₂.

Thus, the potential energy of the system
= the total work done in assembling the configuration