Potential due to an Electric Dipole

The potential due to an electric dipole at a point in space is the electric potential energy per unit charge that a test charge would experience at that point due to the dipole. An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside an electric field without producing acceleration. In this article, we will discuss the potential due to an electric dipole and its derivation.

Electric Potential

Electric potential describes the amount of electric potential energy per unit charge at a point in space. It is measured in volts. It represents the work needed to move a positive electric charge from a reference point to a specific point within the field, without producing any acceleration. It is measured in volts. It indicates how much potential energy a unit charge would gain or lose moving into that point in the field.

Electric Dipole and Dipole Moment

An electric dipole is a pair of charges of equal magnitude but opposite signs , separated by a small distance. Theoretically, an electric dipole is defined by the first-order term of the multipole expansion. It consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.

The dipole moment is a measure of the polarity of a molecule. It is defined as the product of the magnitude of the partial charges (q) and the distance (d) between them. The SI unit for electric dipole moment is coulomb meter (C⋅m). The another unit is debye (D) is

Potential due to an Electric Dipole

The electric potential is the work required to move a unit of positive charge from a reference point to a particular point within an electric field having no acceleration. A dipole is referred to as a pair of opposite charges having equal magnitudes that are separated by a distance, d.

Electric potential (V) at a point due to an electric dipole is given by the following expression:

V = k.p.cos θ/ r²

where

• [Tex]k = 1/4\pi\epsilon_\omicron[/Tex], k is Coulomb’s constant
• p is the magnitude of electric dipole, given by p = q.d
• θ is the angle between the dipole moment vector p and the position vector, r .

This formula tells us that the electric potential due to an electric dipole decreases with the square of the distance r and depends on the angle between the dipole moment and the position vector.

Proof of Potential due to an Electric Dipole

Let us consider an electric dipole consist of two equal and opposite point charges –q at A and +q at B, separated by a small distance AB = 2a, with center at O.

Dipole moment, p = q×2a

We will calculate potential at any point P, where

OP = r and ∠BOP = θ

Let BP = r₁ and AP = r₂

Draw AC perpendicular PQ and BD perpendicular PO

In ΔAOC cos θ = OC/OA = OC/a

OC = acosθ

Similarly, OD = acosθ

Potential at P due to +q = 1/4πϵ₀.qr₂

Potential at P due to -q = 1/4πϵ₀.qr₁

Net potential at P due to the dipole

V = 1/4πϵ₀(q/r₂ − q/r₁)

V = q/4πϵ₀(1/r₂ − 1/r₁)

Now, r₁ = AP = CP

r₁ = OP + OC

 r₁ = r + acosθ

And r₂ = BP = DP

r₂ = OP – OD

r₂ = r – acosθ

[Tex]V = \frac{q}{4\pi \epsilon_{0} }(\frac{1}{r – a .cos\theta }-\frac{1}{r + a.cos\theta})[/Tex]

[Tex]V = \frac{q}{4\pi \epsilon_{0} }(\frac{2.a.cos\theta}{r^2 – a^2.cos^2\theta })[/Tex]

[Tex]V = k(\frac{p.cos\theta}{r^2 – a^2.cos^2\theta })[/Tex]

where,

• k is Coulomb Constant and is given as [Tex]k = 1/4\pi\epsilon_\omicron[/Tex]
• p is dipole moment given as p = 2aq

Special Cases

Case 1: When the point P lies on the axial line of the dipole, θ=0^∘ , cosθ = 1

V = p/r²−a²

If a<<r ⇒ V= p/r²

Thus, due to an electric dipole, potential, V∝ 1/r²

Case 2: When the point P lies on the equatorial line of the dipole, θ = 90^∘, cosθ = 0

This means electric potential due to an electric dipole is zero at every point on the equatorial line of the dipole.

This expression provides a mathematical description of how the electric potential varies around an electric dipole. It is fundamental in understanding the behavior of electric fields and potentials in dipole systems.

Conclusion

This expression provides a mathematical description of how the electric potential varies around an electric dipole and is fundamental in understanding the behavior of electric fields and potentials in dipole systems.

Dependence on Distance : The potential decreases with the square of the distance from the dipole. This means that as you move away from the dipole, the potential decreases rapidly, following an inverse square law similar to that of the electric field.

Directional Dependency: The potential is also dependent on the angle between the position vector and the dipole moment vector. This angular dependency is described by the cosine function, indicating that the potential is maximum along the axis passing through the midpoint of the dipole and decreases as the angle between the position vector and the dipole moment vector increases.

Symmetry: The potential due to an electric dipole exhibits a certain degree of symmetry. It is symmetric with respect to the axis passing through the midpoint of the dipole, meaning that the potential has the same magnitude on opposite sides of this axis.

Relationship with Electric Field: The electric potential due to a dipole is related to the electric field it generates. The electric field is the negative gradient of the electric potential. This relationship allows for a deeper understanding of the interplay between electric fields and potentials in dipole systems.

In summary, the electric potential due to an electric dipole is a fundamental concept in electromagnetism, providing valuable insights into the behavior of electric fields and potentials in dipole systems and finding applications in a wide range of scientific and technological fields.

FAQs on Potential due to an Electric Dipole

What is an electric dipole?

An electric dipole consists of two equal and opposite charges separated by a distance. It is a fundamental concept in electromagnetism and arises in various physical situations, such as the separation of positive and negative charges within a molecule.

What is electric potential due to an electric dipole?

The electric potential due to an electric dipole at a point in space is the scalar quantity that represents the electrical potential energy per unit charge at that point, produced by the electric dipole.

How to calculate electric potential due to a dipole?

The electric potential due to an electric dipole can be calculated using the principle of superposition, considering the potentials produced by each charge in the dipole and summing them. The resulting expression involves the distance from the charges to the point where the potential is being calculated.

Where is the electric potential due to an electric dipole zero?

The electric potential due to an electric dipole is zero at points located along the perpendicular bisector of the dipole, equidistant from its positive and negative charges. This line is known as the equatorial line of the dipole.

What are the applications of the electric potential due to an electric dipole?

The electric potential due to an electric dipole is fundamental to understanding various phenomena in electromagnetism, such as the behavior of molecules, interactions between charges, and the operation of devices like capacitors and antennas. It also plays a role in applications such as molecular modeling, electrostatics, and electrical engineering.