JEE Main & Advanced Physics Electrostatics & Capacitance Electric Dipole

System of two equal and opposite charges separated by a small fixed distance is called a dipole.

(1) Dipole moment : It is a vector quantity and is directed from negative charge to positive charge along the axis. It is denoted as \[\vec{p}\] and is defined as the product of the magnitude of either of the charge and the dipole length  i.e.  \[\vec{p}=q\,(2\vec{l})\]

Its S.I. unit is coulomb-metre or Debye (1 Debye \[=3.3\times {{10}^{-30}}\,C\times m\] ) and its dimensions are \[{{M}^{0}}{{L}^{1}}{{T}^{1}}{{A}^{1}}\].

(2) When a dielectric is placed in an electric field, its atoms or molecules are considered as tiny dipoles.

Water \[({{H}_{2}}O),\] Chloroform \[(CHC{{l}_{3}}),\] Ammonia \[(N{{H}_{3}}),\,HCl,\,CO\] molecules are some example of permanent electric dipole.

(3) Electric field and potential due to an electric dipole : If a, e and g are three points on axial, equatorial and general position at a distance r from the centre of dipole

(i) At axial point : Electric field and potential are given as

\[{{E}_{a}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{2p}{{{r}^{3}}}\] (directed from \[-q\] to +q)

\[{{V}_{a}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{p}{{{r}^{2}}}\].  Angle between \[{{\overrightarrow{E}}_{a}}\] and \[\overrightarrow{p\,}\] is \[{{0}^{o}}\].

(ii) At equatorial point : \[{{E}_{e}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{p}{{{r}^{3}}}\] (directed from +q to \[-q\]) and \[{{V}_{e}}=0\]. Angle between \[{{\overrightarrow{E}}_{e}}\] and \[\vec{p}\] is \[{{180}^{o}}\].

(iii) At general point : \[{{E}_{g}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{p}{{{r}^{3}}}\sqrt{(3{{\cos }^{2}}\theta +1)}\] and \[{{V}_{g}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{p\cos \theta }{{{r}^{2}}}\]. Angle between \[\overrightarrow{E}\] and \[\overrightarrow{p}\] is \[(\theta +\alpha )\](where \[\tan \alpha =\frac{1}{2}\tan \theta \])

(4) Dipole in an external electric field : When a dipole is kept in an uniform electric field. The net force experienced by the dipole is zero as shown in fig.

The net torque experienced by the dipole is

\[\tau =pE\sin \theta \]

\[\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}\]

Hence due to torque so produced, dipole align itself in the direction of electric field. This is the position of stable equilibrium of dipole.

(i) Work done in rotation : Suppose initially, dipole is kept in a uniform electric field at an angle \[{{\theta }_{1}}\]. Now to turn it through an angle \[{{\theta }_{2}}\] (with the field) Work done  \[W=pE(\cos {{\theta }_{1}}-\cos {{\theta }_{2}})\].

If \[{{\theta }_{1}}={{0}^{o}}\] and \[{{\theta }_{2}}=\theta \] i.e. initially dipole is kept along the field then it turn through \[\theta \] so work done \[W=pE(1-\cos \theta )\]

(ii) Potential energy of dipole : It is defined as work done in rotating a dipole from a direction perpendicular to the field to the given direction, i.e. from above formula of work.

If \[{{\theta }_{1}}={{90}^{o}}\] and \[{{\theta }_{2}}=\theta \Rightarrow W=U-pE\cos \theta \]

\[\theta ={{0}^{o}}\]	\[\theta ={{90}^{o}}\]	\[\theta ={{180}^{o}}\]
Stable equilibrium	Not in equilibrium	Unstable equilibrium
\[\tau =0\]	\[{{\tau }_{\max }}=pE\]	\[\tau =0\]
\[W=0\]	\[W=pE\]	\[{{W}_{\max }}=2pE\]
\[{{U}_{\min }}=-pE\]	\[U=0\]	\[{{U}_{\max }}=pE\]

(iii) Equilibrium of dipole : When \[\theta ={{0}^{o}}\] i.e. dipole is placed along the electric field it is said to be in stable equilibrium, because after turning it through a small angle, dipole tries to align itself again in the direction of electric field.

When \[\theta ={{180}^{o}}\] i.e. dipole is placed opposite to electric field, it is said to be in unstable equilibrium.

(iv) Oscillation of dipole : In a uniform electric field if a dipole is slightly displaced from it's stable equilibrium position it executes angular SHM having period of oscillation.

\[T=2\pi \sqrt{\frac{I}{pE}}\]  where \[l=\] moment of inertia of dipole about the axis passing through it's centre and perpendicular to it's length.

(5) Electric dipole in non-uniform electric field : In non-uniform electric field \[{{F}_{net}}\ne 0,\,{{\tau }_{net}}\ne 0\]

Motion of the dipole is combination of translatory and rotatory motion

Dipole-dipole interaction

Relative position of dipole	Force	Potential energy
	\[\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{6{{p}_{1}}{{p}_{2}}}{{{r}^{4}}}\] (attractive)	\[\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{2{{p}_{1}}{{p}_{2}}}{{{r}^{3}}}\]
	\[\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{3{{p}_{1}}{{p}_{2}}}{{{r}^{4}}}\] (repulsive)	\[\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{p}_{1}}{{p}_{2}}}{{{r}^{3}}}\]
	\[\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{3{{p}_{1}}{{p}_{2}}}{{{r}^{4}}}\] (perpendicular to r )	0