Define Equipotential Surface , It's Type and Properties - Param Himalaya

EQUIPOTENTIAL (SAME POTENTIAL) SURFACE 

Define equipotential surface. List the factors on which formation of equipotential surface depends. Draw and explain equipotential surfaces for (i) uniform electric field, (ii) isolated point charge, (iii) a pair of similar point charges and (iv) electric dipole.

Equipotential Surface: A surface whose every point has same electric potential due to charge configuration is called equipotential surface.

An equipotential surface is defined as the locus of all the points in a medium at which electric potential due to a charge configuration is same.

The formation of an equipotential surface will depend upon the type of medium i.e., isotropic or non-isotropic and the amount of charge distribution.

1. Equipotential surfaces for a uniform electric field

Uniform electric field is represented by equidistant parallel straight lines. Draw planes I, II and III perpendicular to the direction of electric field (Figure 17). The potential $V_1$ at every point on the plane I is same. So plane I is equipotential surface. Similarly, potential $V_2$ is same at every point on the plane II, so the plane II is also equipotential surface. Similarly, potential $V_3$ is same at every point on the plane III, therefore, the plane III is equipotential surface. Thus, equipotential surfaces for an uniform electric field are planes perpendicular to the electric field lines representing uniform electric field.

2. Equipotential surfaces for an isolated point charge

Potential due to an isolated point charge + q at a distance r is given by

$V = \frac{q}{4\pi\epsilon_0 r}$

If we draw a sphere of radius r around + q charge, then all the points on this sphere will have the same potential. Thus, equipotential surfaces for an isolated point charge are concentric spherical surfaces around the isolated point charge

The potential on a spherical surface goes on decreasing as the radius of spherical surface increases ($V_1 > V_2 > V_3$).

3. Equipotential surfaces for a pair of similar point charges 

The shape of equipotential surface depends upon the algebraic sum of potentials at different point due to each charge. Figure shows the equipotential surfaces for two positive charges of same value.

4. Equipotential surfaces for an electric  dipole. 

Properties of Equipotential Surfaces

Write and explain various properties of equipotential surfaces.

1. No work is done in moving a test charge from one point to another point on an equipotential surface.

Consider two points A and B on an equipotential surface. Potential difference between points A and B is given by,

$V_B - V_A = \frac{W_{AB}}{q_0} \quad \text{or} \quad W_{AB} = q_0 (V_B - V_A)$

Since every point on the equipotential surface has the same value of the potential i.e., $V_A = V_B$

$W_{AB} = q_0 \times 0 = 0$

Thus, no work is done in moving a test charge from one point to another point on the equipotential surface.

2. The electric field is perpendicular to the equipotential surface.

We know, $\underline{dW = \vec{E} \cdot d\vec{r}}$

As no work is done in moving a test charge on the equipotential surface i.e., $dW = 0$

$\vec{E} \cdot d\vec{r} = 0 \quad \text{or} \quad E dr \cos \theta = 0$

$\cos \theta = 0 \quad \text{or} \quad \theta = 90^\circ$

Thus, $\vec{E}$ is perpendicular to $d\vec{r}$.

3. Equipotential surfaces indicate regions of strong or weak electric fields.

Using,

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

or

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

 Since $dV$ (i.e. potential difference) is constant on the equipotential surface, so

$dr \propto \frac{1}{E}$

If $E$ is strong (i.e. large), $dr$ will be small i.e. the separation of equipotential surfaces will be smaller. Thus, equipotential surfaces are closer in the region of strong electric field.

On the other hand, equipotential surfaces are far apart in the region of weak electric field.

4. Two equipotential surfaces cannot intersect with each other.

If two equipotential surfaces intersect each other, then at the points of intersection, there will be two values of the electric potential due to a point charge. This is not possible. Hence, two equipotential surfaces can not intersect.