A surface on which the potential is the same everywhere is called equipotential surface.
Characteristics of equipotential surfaces
1. No work is done in moving a test charge over an equipotential surface,
Since potential difference = \(\frac{Work \ done}{ Charge }\)
For an equipotential surface,
potential difference = 0
∴ \(\frac{Work \ done}{ Charge }\) = 0
or Work done = 0
2. Electric field intensity \(\vec E \perp\) to the equipotential surface.
Since potential difference between two points A and B = -line integral of electric field
i.e. dV = \(-\int_A^B \vec E. \vec{dl}\)
Since for an equipotential surface, dV = 0 B
∴ \(\int_A^B \vec E. \vec{dl}\) = 0 ......................(i)
i.e. line integral of electric field between any two points on equipotential surface is zero.
From eq. (i), \(\vec E. \vec{dl}\) = 0
or \(\vec E\) ⊥ \(\vec{dl}\)
3. Equipotential surface helps us to distinguish regions of strong fields from those of weak fields.
So equipotential surfaces are closer together, where the electric field is stronger and vice versa as shown in Fig.
Example
(i) In the case of an isolated point charge, all points equidistant from the charge are at the same potential. Thus, for a point charge, equipotential surfaces will be a series of concentric spherical shells with the charges at center Fig. The potential at each of the spherical surfaces (I, II and III) is different but for all points on any one surface, it is the same. Since,
E = \(-\frac{dV}{dr}\) (for constant dV)
dr ∝ \(\frac{1}{E}\).
It follows that for the same charge in potential, distance between equipotential surfaces increases with decrease in electric field.
(ii) For a uniform electric field, the equipotential surfaces are at right angles to the field lines Fig.
(iii) Fig. and show respectively the equipotential surfaces and electric lines of force for two equal positive charges and a pair of equal and opposite charges. The dotted lines
represent electric lines of force. The thick circles around the charges represent equipotential surfaces due to individual charges. The equipotential surfaces due to charges is obtained by adding potentials at a point due to two charges algebraically. They have been represented by thick curves or lines.
