Dipole Electric Field

The electric dipole moment is the product of either of two charges (ignoring the sign) and the distance between them.

A dipole is an arrangement of two charges bearing the same magnitude but an opposite polarity separated by some distance. 

So, if there are two charges and we join the center of these two charges with an imaginary line and the distance between them is ‘2a’, then the dipole moment is: 

\[\vec{P} = q(2\vec{a})\]

Here,

p = electric dipole moment, and it has a direction, i.e., a vector quantity

q = charge

2a = dipole length (a vector quantity) = displacement of - q charge w.r.t. + q. 

Electric Field due to Electric Dipole 

The study of electric dipoles is important for an electrical phenomenon in the matter. We know that a matter contains atoms and molecules, and each has positively charged and negatively charged nuclei. If the center of the mass of the positive nuclei coincides with the negative nuclei, it possesses an internal or permanent dipole moment. 

In the absence of an electric field, the dipole moments are randomly oriented such that the net dipole moment of the system becomes zero. 

When an electric field is supplied to the system of charges inside the matter, the polar molecules align themselves in the direction of the electric field, and some net dipole moment develops, and the matter is said to be polarized. 

So, the field of an electric dipole is the space around the dipole which can be experienced by the effect of an electric dipole, so let’s discuss the electric field due to the dipole. 

Electric Field due to Dipole at any Point.

Let’s take an arrangement for charges viz: electric dipole, and consider any point on the dipole. 

Let there be a system of two charges bearing + q and - q charges separated by some distance ‘2a’, and how to calculate the electric field of a dipole. 

Since the distance between the center of the dipole length and the point P is ‘r’ and the angle made by the line joining P to the center of the dipole is θ. 

We know that the electric field due to dipole is: 

On Axial Line of Electric Dipole

\[\vec{|E|} = \frac{\vec{|P|}}{4\pi \epsilon_{o}} . \frac{2r}{(r^{2}-a^{2})^{2}}\]

If the dipole length is short, then 2a<<r,

so the formula becomes:

\[\vec{|E|} = \frac{\vec{|P|}}{4\pi \epsilon_{o}} . \frac{2}{r^{3}}\]

On Equatorial Line of Electric Dipole

The formula for the equatorial line of electric dipole is:

\[\vec{|E|} = \frac{\vec{|P|}}{4\pi \epsilon_{o}} . \frac{2r}{(r^{2}+a^{2})^{2}}\]

If the dipole is short, the formula becomes:

\[\vec{|E|} = \frac{\vec{|P|}}{4\pi \epsilon_{o}} . \frac{2}{r^{3}}\]

Let ‘O’ be the center of the dipole and consider point ‘P’ lying on the axial line of the dipole, which is at distance ‘r’ from the center ‘O’ such that OP = r.

p \[cos\theta\] is along \[A_{1} B_{1}\] and p \[sin\theta\] is along \[A_{1} B_{1} \perp A_{2} B_{2}\].

So, the electric field intensity will be:

\[|\vec{E_{1}}| = \frac{2p\cos\theta}{4\pi \epsilon_{o}} .\frac{1}{r^{3}}\]

Let it be represented by \[\vec{KL}\] along  with \[\vec{OK}\], and the field intensity at k will be:

\[|\vec{E_{2}}| = \frac{2p\sin\theta}{4\pi \epsilon_{o}} .\frac{1}{r^{3}}\]

Let it be represented by \[\vec{KM}\] parallel to \[B_{2} A_{2}\],

and perpendicular to \[\vec{KL}\]. 

Complete the rectangle KLNM, and join \[\vec{KN}\].

Now, applying the 2nd law of vector addition, \[\vec{KN}\] represents the resultant electric field,

which is given by:

KN = \[sqrt{KL^{2} + KM^{2}}\]

   = \[sqrt{E_1^2 +E_2^2}\]

   = \[sqrt{(\frac{2p\cos\theta}{4\pi \epsilon_{o}} .\frac{1}{r^{3}})^{2} + (\frac{2p\sin\theta}{4\pi \epsilon_{o}} .\frac{1}{r^{3}})^{2}}\]

   = \[\frac{P}{4\pi \epsilon_{o}r^{3}}\sqrt{4\cos^{2}\theta + \sin^{2}\theta}\]

\[\vec{|E|} = \frac{P}{4\pi \epsilon_{o}r^{3}}\sqrt{3\cos^{2}\theta + 1}\]......(3)

So, we get the electric field of a dipole in eq(3)

Also, let LKN = \[\beta\], then \[\triangle\]KLN is:

\[tan\beta = \frac{LN}{KL} = \frac{KM}{KL}\]

= \[(\frac{p\sin\theta}{4\pi \epsilon_{o}} .\frac{1}{r^{3}})^{2} \times \frac{4\pi \epsilon_{o}}{2p\cos\theta}.\frac{1}{r^{3}}\]

\[tan\beta = \frac{1}{2} \tan\theta\] .....(4)

Now, here we will consider two cases viz: Field along the axial line of the dipole and the second one for the field along the equatorial line of the dipole.

1. When Point K Lies Along the Axial Line of Dipole.At this moment, θ = 0° = Cos 0° = 1Now, equation (3) becomes:

\[\vec{|E|} = \frac{P}{4\pi \epsilon_{o}r^{3}}\sqrt{3\cos^{2}0^{0} + 1}\]

           \[ = \frac{2P}{4\pi \epsilon_{o}r^{3}}\]

And, \[tan\beta = \frac{1}{2} \tan 0^{0}\]

\[=\beta = 0^{0}\]

This shows that the electric field intensity is along the axial line of the electric dipole.

2. When the point K lies on the equatorial line of the dipole.At this moment, θ = 90° = Cos 90° = 0 From eq (3), we get:

\[frac{P}{4\pi \epsilon_{o}r^{3}}\sqrt{3\cos^{2}90^{0} + 1}\]

\[\vec{|E|} = \frac{P}{\epsilon_{o}r^{3}}\]

And, \[tan 90^{0} = \frac{1}{2}\tan\theta\]

\[= \frac{1}{2}\tan 90^{0} = \infty\]

\[= \beta = \tan 90^{0}\]

Here, the angle 90° shows that the direction of the resultant electric field intensity is perpendicular to the equatorial line, and therefore, parallel to the axial line of a dipole.

Dipole electric field is a part of physics and it is discussed in detail in Chapter 1 electric charges and fields of the NCERT book of Class 12. It is considered an extremely important concept as it is prescribed by the Central board of secondary education that is the CBSE. It carries significant weightage in the Class 12 board examination and therefore it is advisable for students to study this chapter and this concept called Dipole electric field in depth.

To make the learning process easier and fun the Vedantu’s team of expert teachers who have done extensive research and have years of experience in the concerned field have curated the study material that is based on the CBSE curriculum. This article on dipole electric fields is written in an extremely simplified manner as the objective of writing this is to help students understand and get a good hold of the concept of electric charges and fields.

Along with the study material Vedantu teachers have also provided students with practice questions with their solutions so that students can keep in check their progress and can get to know about their strengths and weaknesses.

The study notes on the dipole electric field can be easily accessed by visiting Vedantu’s website. Vedantu’s team has provided the study material in a PDF format which makes it even more convenient for students as it can be for free and can be used anywhere especially in an offline environment.

This chapter mainly deals with electrostatics which is basically the study of forces, fields, and potentials that arise from static charge.

Fun Fact-

The concept of electricity was first discovered by Thales of Miletus, Greece, around 600 BC when he delved into how amber when rubbed with wool or silk cloth attracts light objects. The term electricity originated from the greek word called elektron which means amber. During that time various metals were discovered which when dropped could attract light objects like straws, bits of paper, or hair.

Key Concepts Needed to Understand Dipole Electric Field are-

1.1 ELECTRIC CHARGE

1.2 CONDUCTORS AND INSULATORS

1.3 CHARGING BY INDUCTION

1.4 BASIC PROPERTIES OF ELECTRIC CHARGE

1.4.1 Additivity of charges

1.4.2 Charge is conserved

1.4.3 Quantisation of charge

1.5 COULOMB’S LAW

1.6 FORCES BETWEEN MULTIPLE CHARGES

1.7 ELECTRIC FIELD

1.7.1 Electric field due to a system of charges

1.7.2 Physical significance of electric field

1.8 ELECTRIC FIELD LINES

1.9 ELECTRIC FLUX

1.10 ELECTRIC DIPOLE

1.10.1 The field of an electric dipole

1.10.2 Physical significance of dipoles

1.11 DIPOLE IN A UNIFORM EXTERNAL FIELD

1.12 CONTINUOUS CHARGE DISTRIBUTION

1.13 GAUSS’S LAW

1.14 APPLICATIONS OF GAUSS’S LAW

1.14.1 Field due to an infinitely long straight uniformly charged wire

1.14.2 Field due to a uniformly charged infinite plane sheet

1.14.3 Field due to a uniformly charged thin spherical shell