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Equipotential Surfaces

The electric field is defined as an electric force divided per unit charge. The direction of the electric field is in the direction of the electric force which is exerted on the electrically charged of electric fields. The electric field is exerted in the outside direction on a positive charge while it is in the downward side on negative point charge. This is measured by coulomb’s law. When the positive charge is moved in the direction of electric field then this loose the potential energy while the movement of positive charge in the opposite direction of an electric field is increased the potential energy of the object.

The electric potential is related to the potential energy of a positive charge in an electric field. But if there is any surface which contains constant electric potential then this is an equipotential surface. Or we can say that the value of potential difference of two point charges is equal to zero on an equipotential surface. Let’s discuss about equipotential surface and lines, how these can be drawn, and its properties.

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Electric Fields and Equipotential Surfaces

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As we discussed above, an equipotential surface is the locus of points at which the potential due to a charge distribution is the same.

The potential at any point in an electric field is given by

$V$ = $\frac{1}{4 \pi \varepsilon_{0}}\frac{q}{r}$

Where $q$ is the source charge and $r$ is the distance of the observation point from the source charge.
Equipotential surfaces are more crowded in a region of strong electric field than in the region of weak electric field. If $dV$ is the potential spacing between two equipotential surfaces and $dx$ is the perpendicular distance between two surfaces under consideration, then electric intensity is given by

$E$ = $-\frac{dV}{dx}$

If $dV$ is fixed, then $E \propto$ $\frac{1}{dx}$ or $dx \propto$ $\frac{1}{E}$.

As the field become stronger, $E$ increases and hence $dx$ decreases. This explains the crowding of equipotential surfaces in a region of strong electric field.

An Equipotential Surface Must be?

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Properties of Equipotential Surface 

Electric Field

1. If a test charge is moved from one equipotential surface say $V_{A}$ to another equipotential surface, say $V_{B}$, work done is equal to

$W = q_{0}(V_{B}-V_{A})$ 

where $q_{0}$ is the test charge. On the same surface, $V_{A} = V_{B}$, so work done = $q_{0}(V_{A}-V_{A})$ = $0$. No work is done in moving a test charge from one point on an equipotential surface to another point on the same equipotential surface.

2) The electric field lines are perpendicular to the equipotential surfaces. 

3) Equipotential surfaces indicates region of strong or weak electric fields. Thus in strong electric field, the equipotential surfaces are closely spaced while in weak electric fields, equipotential surfaces are widely spaced.

4) Two equipotential surfaces cannot intersect. If they intersect, then at the line of intersection, there will be two values of  potential, which is not possible. However, inside an equipotential space two equipotential surface can intersect.      

Equipotential Surfaces in a Capacitor

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Consider equipotential surfaces between the plates of a parallel plate capacitor. If you start at the negatively charged plate and move a distance $x$ toward the positively charged plates, we have 

$V$ = $\frac{qx}{\varepsilon_{0} A}$     

In other words, the equipotential surface here depend only on how far the point between the two plates. The below figure which shows two equipotential surfaces between the plates of the parallel plate capacitor. This is analogous to the gravitational potential close to the ground, which increase in proportion to height.       


Equipotential Lines

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The points in an electric field, which are at the same electric potential, are called equipotential points. If such points all be connected by a line (or curve), it is called an equipotential line. If such points all lie on a surface, it is called an equipotential surface. If such points are distributed throughout some space or volume, it is called an equipotential volume.

Drawing Equipotential Lines

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1. The uniform electric produced by an infinite sheet of charge : In a uniform electric field, the equipotential surfaces are planes parallel to each other and perpendicular to the field lines.  

Infinite Sheet

2. A spherically symmetric electric field produced by a point charge: The  equipotential surfaces in the field of a point charge are concentric spheres,with the center on the point charge.

Point Charge

3. An electric field produced by an electric dipole: These are elliptically shaped with semi major axis parallel to equatorial line and semi minor axis along the axial line. At the center of the dipole, the equatorial line becomes the equipotential line.


Equipotential Lines and Electric Field

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If electric field lines are perpendicular to an equipotential surface, then conversely the equipotential surface must be perpendicular to the field lines. Thus equipotential surfaces are always perpendicular to the direction of the electric field $\vec{E}$, which is tangent to the field lines.
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