Equipotential Surfaces | Electrostatic Potential and Capacitance | Notes | Summary - Zigya

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Electrostatic Potential and Capacitance

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Equipotential Surfaces

An equipotential surface is a surface with a constant value of the potential at all points on the surface.  For a single charge q, the potential is given
by, straight V space equals space fraction numerator 1 over denominator 4 πε subscript 0 end fraction straight q over straight r
This shows that V is a constant if r is constant. Thus, equipotential surfaces of a single point charge are concentric spherical surfaces centre at the charge.

No work in done in moving from one point to another in the equipotential surface.

For a uniform electric field E, say, along with the x-axis, the equipotential surfaces are planes normal to the x-axis, i.e., planes parallel to the y-z plane.

Relation between field and potential

(i) The electric field is in the direction in which the potential decreases steepest.
(ii) Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.

Consider two closely spaced equipotential surfaces A and B Fig. with potential values V and V + δV, where δV is the change in V in the direction of the electric field E.

Let P be a point on the surface B. δl is the perpendicular distance of the surface A from P. Imagine that a unit positive charge is moved along this perpendicular from the surface B to surface A against the electric field. The work done in this process is |E|δl.

This work equals the potential difference VA –VB. Thus,

Thus comma
vertical line straight E vertical line straight delta l space equals space straight V space minus space left parenthesis straight V space plus space δV right parenthesis space equals space minus δV
straight i. straight e. comma space vertical line straight E vertical line space equals space minus space fraction numerator straight delta space straight V over denominator straight delta l end fraction
Since space δV space is space negative space δV space equals space minus space vertical line δV vertical line comma space
rewrite
vertical line straight E vertical line space equals space minus space fraction numerator δV over denominator straight delta l end fraction space equals space plus fraction numerator vertical line straight delta space straight V vertical line over denominator straight delta l end fraction

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