# Dipole Electric Field

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## Electric Dipole Moment

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:

$\overrightarrow{p} = q(2\overrightarrow{a})$

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 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

$|\overrightarrow{E}| = \frac{|\overrightarrow{P}|}{4 \pi \epsilon_{0}} \cdot \frac{2r}{(r^{2} - a^{2})^{2}}$

If the dipole length is short, then 2a << r, so the formula becomes:

$|\overrightarrow{E}| = \frac{\overrightarrow{P}|}{4 \pi \epsilon_{0}} \cdot \frac{2}{r^{3}}$

• On Equatorial Line of Electric Dipole

The formula for the equatorial line of electric dipole is:

$|\overrightarrow{E}| = \frac{|\overrightarrow{P}|}{4 \pi \epsilon_{0}} \cdot \frac{2r}{(r^{2} + a^{2})^{2}}$

If the dipole is short, the formula becomes:

$|\overrightarrow{E}| = \frac{\overrightarrow{P}|}{4 \pi \epsilon_{0}} \cdot \frac{2}{r^{3}}$

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 Cosθ is along A1B1 and p Sinθ is along A1B1 ⊥ A2B2. So, the electric field intensity will be:

$|\overrightarrow{E_{1}}| = \frac{2p cos \theta}{4 \pi \epsilon_{0}} \cdot \frac{1}{r^{3}}$

Let it be represented by $\overrightarrow{KL}$ along with $\overrightarrow{OK}$, and the field intensity at K will be:

$|\overrightarrow{E_{2}}| = \frac{2p sin \theta}{4 \pi \epsilon_{0}} \cdot \frac{1}{r^{3}}$

Let it be represented by $\overrightarrow{KM}$ parallel to B2A2, and perpendicular to $\overrightarrow{KL}$. Complete the rectangle KLNM, and join $\overrightarrow{KN}$.

Now, applying the llgm law of vector addition, $\overrightarrow{KN}$ represents the resultant electric field, which is given by:

$KN = \sqrt{KL^{2} + KM^{2}}$

$= \sqrt{E_{1}^{2} + E_{2}^{2}}$

$= \sqrt{(\frac{2p cos \theta}{4 \pi \epsilon_{0}} \cdot \frac{1}{r^{3}})^{2} + (\frac{2p sin \theta}{4 \pi \epsilon_{0}} \cdot \frac{1}{r^{3}})^{2}}$

$= \frac{p}{4 \pi \epsilon_{0} r^{3}} \sqrt{4 cos^{2} \theta + sin^{2} \theta}$

$|\overrightarrow{E}| = \frac{p}{4 \pi \epsilon_{0} r^{3}} \sqrt{3 cos^{2} \theta + 1}$. . . . . .(3)

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

Also, let LKN = β, then ΔKLN is:

tanβ = LN/KL = KM/KL = $(\frac{p sin \theta}{4 \pi \epsilon_{0}} \cdot \frac{1}{r^{3}}) \times \frac{4 \pi \epsilon_{0}}{2p cos \theta} \cdot \frac{1}{r^{3}}$

tanβ = ½ tanθ. . . . .(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° = 1

Now, equation (3) becomes:

$|\overrightarrow{E}| = \frac{p}{4 \pi \epsilon_{0} r^{3}} \sqrt{3 cos^{2} 0^{\circ} + 1} = \frac{2p}{4 \pi \epsilon_{0} r^{3}}$

And,   tan β = ½ tan 0°

= β = 0°

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

1. 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:

$\frac{p}{4 \pi \epsilon_{0} r^{3}} \sqrt{3 cos^{2} 90^{\circ} + 1}$

$|\overrightarrow{E}| = \frac{p}{\epsilon_{0} r^{3}}$

And,   tan 90° = ½ tan θ

= ½ tan 90° = ∞

= β = 90°

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.

Question 1: What do you Mean by the Field of an Electric Dipole?

Answer: We know that when a system is subjected to an external field, the net dipole moment generates in its direction.

Therefore, the space around it in which the electric effect of the dipole can be experienced is called the field of an electric dipole or the dipole field.

Question 2: An Electric Dipole is Placed at Rest in a Uniform Electric Field, and Released. How will it Move?

Answer: When an electric dipole is placed in a uniform electric field, a torque develops and aligns the dipole in the direction of an electric field.  However, the dipole doesn’t move as the net force acting on the dipole is zero.

Question 3: What is the Unit of the Electric Dipole Moment?

Answer: In international systems, the unit of the dipole moment is Coulomb-meter or C-m. However, two more units are commonly used for the same. These are:

• StatC. Cm, and

• Debye (D)

Where,

1 StatC. Cm = 3.33564 x 10-30 C.m, and

1 D = 1018 StatC . Cm

Question 4: When is an Electric Dipole in Unstable Equilibrium in an Electric Field?

Answer: Electric dipole is in unstable equilibrium when p is antiparallel to E, i.e., θ = 180°.