From OAC, Â = = i.e., OC = a. Similarly OD = a.

If r>>a, then PA = PC = PO + OC = r + a, and PB = PD = PO – ODÂ = r – a.

If V is the potential at a point P due to electric dipole then,

V = k[]

= k[ – ]

= kq[]

=

Or, V = Â = Â [p = q2a = dipole moment and k = ]

If r>>a, then is neglected. So V = .

Potential energy due to electric dipole:Â

Let us consider an electric dipole consists of -q and +q at points A and B respectively separated by a distance a is placed in the electric field of intensity Â at an angle Î¸.Â The dipole moment is = qa.

The magnitude of torque acting on the dipole is Â = pE.

If dÎ¸ is the angular displacement of the dipole in electric field then the work done stored as the potential energy of the dipole as dw = dÎ¸ = pEdÎ¸. Then the total work done stored into the dipole to rotate it from Î¸_{1} to Î¸_{2} is W = dW = pEÂ = – pE[cos Î¸_{2} – cos Î¸_{1}] = pE[cos Î¸_{1} – cos Î¸_{2}]

If Î¸_{1} = 90^{0} and Î¸_{2} = Î¸ then w = – pEcosÎ¸

1. If Î¸ = 0^{0 }then, W = – pEcos0^{0} = -pE.Â This is stable equilibrium.

2. If Î¸ = 90^{0 }then, w = – pEcos90^{0} = 0

3. If Î¸ = 180^{0 }then, w = – pEcos180^{0} = pE.Â This is unstable equilibrium.

Equipotential surface:

An equipotential surface is defined as the surface, on which all points lying are at the same potential.

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

Electric field is perpendicular to the equipotential surface:

Let us consider A and B are very closed point on the equipotential surface (AB = dr). Let electric field creates angle Î¸ with this surface. EcosÎ¸ is the component of the field along the surface. The potential difference between points A and B is dV = 0. The work done to move a charge q from A to B is dw = q.dV = 0.

Similarly, dw = F.dr = qEcosÎ¸dr = 0. As q, E and dr 0 then cosÎ¸ = 0

Or, cosÎ¸ = cos90^{0} Â

Î¸ = 90^{0} so, electric field is perpendicular to the equipotential surface.

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