# Equipotential Surfaces

All points on an equipotential surface have the same electric potential (i.e., the same charge).

• The electric force neither helps nor hinders motion of an electric charge along an equipotential surface.

• Electric field lines are always perpendicular to an equipotential surface.

• Electric potential is analogous to altitude; one can make maps of each in every similar ways.

An equipotential surface is just what its name declares: a surface-real or imaginary-on which all points have the same electric potential.

(i) All points on a sphere around a point charge have the same distance r, so all have same potential.

V= K e q / r

FIGURE 7.2 Sphere around a point charge

(ii) All points on a plane parallel to the plates a conductor have the same potential

FIGURE 7.3 Plates of conductor

What’s so great about equipotential surfaces?

– the electric field is always perpendicular to an equipotential surface.

FIGURE 7.4 Electric field on equipotential surface

Which implies that a charged particle moving along an equipotential surface always moves perpendicular to the electric force so

-the electric force does no work on particle moving along an equipotential surface.

and conversely, one need not do any work to move the particle i.e., particles may move freely.

The Gravity Analogy (again)

In the earth’s gravitational field, equipotential surfaces are simply flat, level areas: area of constant altitude.

FIGURE 7.5 Gravity Analogy

-the crossed plain is an equipotential; the gravitational force does no work on someone walking on a plain.

-the mountain has a higher gravitational potential; one must do work to climb up it.

-the dotted hole is another equipotential surface.

Topographic maps use lines of constant altitudes (gravitational equipotentials) to show the “shape” of the landscape. (Figure 7.6)

FIGURE 7.6 Topographic maps

One can also make maps the electric potential, using lines of equipotential. (Figure 7.7)

FIGURE 7.7 Maps of Electric Potential                                FIGURE 7.8 (a) Topo map

For small values of radius the equipotential cylindry about each line charge are nearly concentric, with line charges as centre.

On a topo map, lines close together indicate a sharp change in altitude. One can measure the steepness of ground by

Gradient =  Change in altitude   /  distance travelled

On a map of electric field, lines close together indicate a strong electric field. (Figure 7.8 b)

FIGURE 7.8 (b) Closer lines indicates strong electric field

Electric field  =Change in electric potential  / distance travelled

∆v / ∆x  =  Volts/meter.