What is electric dipole in an uniform electric field?

Consider an electric dipole consisting of two charges, -q and +q, separated from each other by a distance 2a and placed in a uniform electric field \(overrightarrow{E}\), in the plane of the page. Let its dipole moment \(overrightarrow{p}\) make an angle θ with \(overrightarrow{E}.\)

(a) Force acting on the dipole

Force acting on charge –q at A, i.e., \(overrightarrow{{{F}_{1}}}=-qoverrightarrow{E}\) and force acting on charge +q at B, i.e., \(overrightarrow{{{F}_{2}}}=+qoverrightarrow{E}\).

Since \(overrightarrow{{{F}_{1}}},and,overrightarrow{{{F}_{2}}}\) are equal in magnitude (qE) but act in opposite directions,

Net force acting on the dipole,

\({{overrightarrow{F}}_{net}}={{overrightarrow{F}}_{1}}+{{overrightarrow{F}}_{2}}=-qoverrightarrow{E}+qoverrightarrow{E}=overrightarrow{0}\)

Thus, net force acting on a dipole in a uniform electric field is zero.

(b) Torque acting on the dipole

Since forces \({{overrightarrow{F}}_{1}},and,{{overrightarrow{F}}_{2}}\) act at different points, they form a couple. The magnitude of the moment of the couple, i.e., torque (t) is given by

t = magnitude of either force × arm of the couple

\(=qEtimes BC=qEtimes 2asin theta =left( 2qa right)Esin theta ,or,tau =pEsin theta\)

Were, p = 2qa

This torque tends to rotate \(overrightarrow{p}\) (hence the dipole) in the direction of \(overrightarrow{E},\) thereby reducing θ and producing clockwise rotation. The direction of this torque (according to right hand rule for the cross-product of two vectors) is into the page [figure (b)] and is represented by the symbol \(otimes\). Since \(overrightarrow{p},and,overrightarrow{E}\) lie in the plane of the page and \(overrightarrow{tau }\) is into the page as shown in figure. (c) eqn. (1) can be generalised to vector form as

\(overrightarrow{tau }=overrightarrow{p}times overrightarrow{E}\) … (2)

Alternative approach:

The expression \(overrightarrow{tau }=overrightarrow{p}times overrightarrow{E}\) can also be obtained as follows.

Torque of \({{overrightarrow{F}}_{1}}\) about O, i.e.,

\({{overrightarrow{tau }}_{1}}=overrightarrow{OA}times {{overrightarrow{F}}_{1}}=overrightarrow{OA}times left( -qoverrightarrow{E} right)=-qleft( overrightarrow{OA}times overrightarrow{E} right)\) \(=qleft( overrightarrow{AO}times overrightarrow{E} right)left( as,,-overrightarrow{OA}=overrightarrow{AO} right)\)

Torque of \(overrightarrow{{{F}_{2}}}\) about O, i.e.,

\({{overrightarrow{tau }}_{2}}=overrightarrow{OB}times {{overrightarrow{F}}_{2}}=overrightarrow{OB}times left( qoverrightarrow{E} right)=qleft( overrightarrow{OB}times overrightarrow{E} right)\)

Net torque acting on the dipole, i.e.,

\(overrightarrow{tau }=overrightarrow{{{tau }_{1}}}+overrightarrow{{{tau }_{2}}}=qleft( overrightarrow{AO}times overrightarrow{E} right)+qleft( overrightarrow{OB}times overrightarrow{E} right)\) \(=qleft( overrightarrow{AO}+overrightarrow{OB} right)times overrightarrow{E}\) \(=qleft( overrightarrow{AB} right)times overrightarrow{E}left( as,overrightarrow{AO}+overrightarrow{OB}=overrightarrow{AB} right)\) \(=overrightarrow{p}times overrightarrow{E}\)

(as \(qtimes AB=qtimes 2a=p\) and the direction of \(overrightarrow{p}\) is from A to B, i.e., along \(overrightarrow{AB}\))

Thus, \(overrightarrow{tau }=overrightarrow{p}times overrightarrow{E}\)

Thus, there acts a torque on a dipole in a uniform electric field tending to align the dipole parallel to the field.

Combining the results of (a) and (b), we conclude that:

In a uniform electric field, a dipole experiences a torque but no net force.

Definition of p

Let θ = 90° and E = 1.

As t = pE sin θ, t = p × 1 × sin 90° = p. Thus:

The electric dipole moment of an electric dipole is numerically equal to the torque required to keep the dipole perpendicular to an electric field of unit intensity.

Special cases:

  1. When θ = 0°, t = 0 and \(overrightarrow{p},and,overrightarrow{E}\) are parallel (to each other) and the dipole is in a position of stable equilibrium.
  2. When θ = 180°, t = 0 and \(overrightarrow{p},and,overrightarrow{E}\) are antiparallel and the dipole is in a position of unstable equilibrium.
  3. When θ = 90°, t = pE sin 90° = pE = maximum. If this value of torque is denoted by \({{tau }_{max }},{{tau }_{max }}=pE,,or,,p=frac{{{tau }_{max }}}{E}.\) Thus, dipole moment of an electric dipole is also defined as the maximum torque acting on it per unit electric field.

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