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.
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.
Table of Content
- Electric Potential
- Electric Dipole
- Potential due to a Dipole at any point
- Derivation of Potential due to an Electric Dipole
- Special Cases