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The term ‘Equipotential Surfaces’ is a compound word that can be expanded as equal potential Surfaces. True to its literal sense, it is the collection of space points with the same electric potential. An equipotential can be a line, a surface or a solid region. The most commonly encountered equipotential is a surface. Hence, we will be paying particular attention to equipotential surfaces.
Electric Potential
We know that energy is a scalar quantity. It is easier to analyse a system with scalar quantities than vector quantities. Hence, electric potential is another tool to analyse systems that interact through Coulomb’s law of attraction.
The electric potential of charge ‘q’ is defined as the amount of work done to bring a positive unit charge from infinity to ‘q’.
Thus, for potential of a positive charge ‘q’ at a radial distance is given a
Where 0 is the permeability in vacuum.
Equipotential Surface
An Equipotential is a spatial region where electric potential has the same value. An Equipotential surface has the same potential on every point that lies on its surface. There can be infinitely many equipotential surfaces.
Thus, for a point charge ‘q’, which has a potential
We notice that it is a function of radial distance. Since the collection of points for which is constant is a spherical surface, the equipotential surface of a point charge will be a series of concentric spherical surfaces.
Work Done along an Equipotential Surface
The work done on moving a particle ‘q’ from point A to point B is related as W = -q(VB – VB) where VA is potential at point A and VB is potential at point B.
On an equipotential surface VA = VB and hence W = 0.
Thus, no work is done on moving a particle when the initial and final points lie on an equipotential.
Electric Field and Equipotential Surfaces
We know that work done dW on moving a charge q through a distance ds is the dot product of Electric Field E and ds.
Thus
dW =qE.ds
For ds on an equipotential surface, we know that dW = 0.
which can only happen if the electric field is zero or perpendicular to ds.
The former cannot be true because the presence of an electric field is why Equipotential surfaces exist in the first place.’
Considering that a unit charge is moved a minute distance l in Electric Field E the Potential difference can be written as
W=El=((V+V)-V)=-V
Thus the magnitude of the electric field (disregarding the minus sign ) is
E=Vl
Properties
No work is done when a particle’s initial and final position is on the same equipotential surface.
Electric fields are always perpendicular to Equipotential surfaces
Equipotential surfaces do not intersect each other.
The direction of the Electric Field is in the order where the decrease in potential is the most.
Conclusion
We analysed that equipotential surfaces can be obtained from given electric field vector space and vice versa. Thus where Coulomb’s law cannot be easily applied, we can receive equipotential surfaces and figure out the corresponding electric field distribution.
We know that point charges give a spherical distribution of equipotential surfaces while charged plates have a planar distribution. Often we approximately draw these surfaces with lines in two-dimensional space. But it is important to note that equipotential surfaces occur in three dimensions.
