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Equipotential Surfaces | What Are Equipotential Surfaces

Equipotential Surfaces | What Are Equipotential Surfaces :- In the following figure if a charge is shifted from point A to B on a surface M which is perpendicular to the direction of electric field, then no work is done because electrostatic force on the charge particle is perpendicular to the direction of it’s displacement.

As no work is done in moving the charge from A to B, we can say that A and B are at the same potentials or we can say that all the points of surface M are at the same potential and hence we call the surface M as “Equipotential Surfaces”.

Definition of Equipotential Surfaces

A surface over which the electric potential remains constant is called an equipotential surface.

Key Properties of Equipotential Surfaces

(1). No Work is Done :

As work done

(2). Perpendicular to Electric Field Lines :

Electric field lines always intersect equipotential surfaces at right angles (90°). This is because electric field is in the direction of maximum decrease of potential.

(3). Closer Surfaces = Stronger Electric Field :

The electric field is stronger where equipotential surfaces are closer. E =−dV/dr , meaning a large change in potential over a small distance implies a strong field.

(4). Equipotential Surfaces Can Take Different Shapes :

(i). For a point charge, they are concentric spherical shells. The potential of a shell of radius r is $\displaystyle V=\frac{kQ}{r}$ .

(ii). For a uniform electric field, they are equally spaced parallel planes perpendicular to the field direction.

(iii). For an electric dipole they are not spherical but curved and symmetrical around the dipole axis. Close to each charge, the equipotential surfaces look like distorted spheres. At the midpoint (Center of Dipole) equipotential surface is a plane perpendicular to the dipole axis with zero potential (because the potentials due to +q and –q cancel out here).

Applications of Equipotential Surfaces

Help in designing electrical equipment to prevent short circuits.
Used in capacitor design, where plates are often considered equipotential surfaces.
Useful in solving electrostatics problems as discussed below :

(a). To find potential difference between two points in a uniform electric field : If we wish to find the potential difference between two points A and B which lie on equipotential surfaces M₁ and M₂ as shown in figure below in a uniform electric field, then it is given by :

V_A – V_B = Ed

(b). To find potential difference between two points in a non – uniform electric field : Suppose there is a liner charge distribution with liner charge density λ C/m. Here we wish to find potential difference between two points A and B.

Consider a general point P at a distance r from the liner charge distribution. The electric field intensity at point P is

$\displaystyle E=\frac{2k\lambda }{r}$

Now the potential difference between points A and B is given by :

$\displaystyle {{V}_{A}}-{{V}_{B}}=\int\limits_{{{r}_{A}}}^{{{r}_{B}}}{Edr}=\int\limits_{{{r}_{A}}}^{{{r}_{B}}}{\frac{2k\lambda }{r}dr}$

$\displaystyle \Rightarrow {{V}_{A}}-{{V}_{B}}=2k\lambda \left[ lo{{g}_{e}}\left( \frac{{{r}_{B}}}{{{r}_{A}}} \right) \right]$

Example 1.

A charge +Q is placed at the centre of a dotted circle.

Work done in taking charge +q from A to B is W₁ and from A to C is W₂ . Then :

(A) W₁ > W₂

(B) W₁ < W₂

(D) W₁ = W₂≠ 0

Solution :

As in conservative force field, work done = change in potential energy (of the charge particle), so from A to B :

W₁ = q(V_B – V_A) …..(i)

Similarly from A to C :

W₂ = q(V_C – V_A) …..(ii)

But V_B = V_C, because both points lie on equipotential surface, so from equations (i) and (ii), we can say W₁ = W₂≠ 0. Hence option (D) is correct.

Example 2.

In diagram four equipotential surfaces are shown. An electron is along the paths 1, 2, 3 and 4. If the work done along the are W₁, W₂, W₃ and W₄ respectively, then their values in eV.

Solution :

As work done = change in potential energy, so

W₁ = qΔV = (-e) (V_f – V_i) = (-e) (10 – 10)

⇒ W₁ = 0

W₂ = qΔV = (-e) (V_f – V_i) = (-e) (20 – 10)

⇒ W₂ = – 10 eV

W₃ = qΔV = (-e) (V_f – V_i) = (-e) (30 – 30)

⇒ W₃ = 0

W₄ = qΔV = (-e) (V_f – V_i) = (-e) (20 – 40)

⇒ W₄ = + 20 eV

Equipotential Surfaces | What Are Equipotential Surfaces

Definition of Equipotential Surfaces

Key Properties of Equipotential Surfaces

Applications of Equipotential Surfaces

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