Electric Dipole

The electric dipole is a fundamental charge configuration in electromagnetics and forms the basis for understanding dielectric materials, molecular polarization, and radiation from antennas. A dipole consists of two equal and opposite point charges separated by a small distance. Although the net charge of a dipole is zero, it produces a non-trivial electric field and potential because the two charges do not completely cancel at finite distances. This configuration appears in polar molecules, dielectric analysis, and as the simplest radiating element in antenna theory.

Core Concept: Dipole Moment

The electric dipole moment p is defined as the product of the charge magnitude Q and the separation distance d between the two charges. It is a vector quantity directed from the negative charge to the positive charge. The SI unit of dipole moment is Coulomb-meter (C.m). In molecular physics, dipole moments are often expressed in Debye units (1 Debye = 3.336 x 10^-30 C.m).

The physical significance of the dipole moment is that it quantifies the degree of charge separation. A larger Q or a larger separation d creates a stronger dipole with more pronounced field effects. Even though the net charge is zero, the dipole moment is not zero, and it is the dipole moment that determines how the configuration interacts with external fields and how strongly it radiates when oscillating.

When a dipole is placed in a uniform external electric field, it experiences no net force (because +Q and -Q forces cancel) but does experience a torque given by tau equals p cross E. This torque tends to align the dipole moment with the external field. The potential energy of the dipole in the field is U equals negative p dot E, which is minimum when p and E are parallel (aligned state) and maximum when they are anti-parallel.

Mathematical Expression: Potential and Field of a Dipole

For a dipole located at the origin with moment p aligned along the z-axis, the potential at a distant point at distance r and polar angle theta (measured from the z-axis) is given by V equals p cos(theta) divided by 4 pi epsilon-naught r squared. This is valid when r is much greater than d (the far-field approximation). The potential falls as 1/r^2, faster than the 1/r dependence of a single point charge, because the two opposite charges partially cancel.

The electric field of the dipole in spherical coordinates has two components. The radial component E_r equals 2p cos(theta) divided by 4 pi epsilon-naught r cubed, and the polar component E_theta equals p sin(theta) divided by 4 pi epsilon-naught r cubed. Both components fall as 1/r^3, faster than the 1/r^2 field of a point charge. This rapid fall-off means dipole fields are more localized than monopole fields, which is why molecular dipole interactions are shorter-range than Coulomb interactions.

Practical Understanding

In dielectric materials, an applied electric field causes slight displacement of positive and negative charge centers in each molecule. This creates a collection of aligned microscopic dipoles throughout the material. The bulk effect is described by the polarization vector P (C/m^2), which is the dipole moment per unit volume. The polarization modifies the effective electric field inside the dielectric, leading to the concept of permittivity and capacitance enhancement in capacitors with dielectric fillings.

In antenna theory, an oscillating electric dipole (the Hertzian dipole) is the simplest radiating element. When the dipole moment varies sinusoidally with time, the time-varying field cannot remain quasi-static and instead propagates outward as an electromagnetic wave. The radiation pattern of a dipole antenna is directly derived from the angular dependence sin(theta) in the dipole field equations, with maximum radiation perpendicular to the dipole axis (theta = 90 degrees) and zero radiation along the axis (theta = 0).

Given:
An electric dipole with Q = 10 x 10^-9 C and d = 0.02 m (p along z-axis).
Find the potential V at point (r = 0.5 m, theta = 60 degrees).

Why this formula applies:
Far-field dipole potential: V = p*cos(theta) / (4*pi*epsilon_0*r^2).
Valid because r = 0.5 m >> d = 0.02 m.

Formula:
V = p * cos(theta) / (4*pi*epsilon_0 * r^2)

Substitution:
p = Q * d = 10e-9 * 0.02 = 2e-10 C.m
cos(60) = 0.5
k = 9e9 N.m^2/C^2
r^2 = 0.25

Calculation:
V = 9e9 * 2e-10 * 0.5 / 0.25
V = 9e9 * 1e-10 / 0.25
V = 0.9 / 0.25
V = 3.6 V

Final Answer: V = 3.6 V at r = 0.5 m, theta = 60 degrees

Exam Tip: Dipole potential falls as 1/r^2 and dipole field falls as 1/r^3. Point charge potential falls as 1/r and field as 1/r^2. Quadrupole potential falls as 1/r^3. GATE may ask you to identify the source (monopole, dipole, or quadrupole) by the power of r in the potential expression.

• Dipole moment p = Qd, direction from -Q to +Q. Unit: C.m.
• Dipole potential: V = p*cos(theta) / (4*pi*epsilon_0*r^2). Falls as 1/r^2.
• Dipole field components both fall as 1/r^3. Compare: point charge potential falls as 1/r, field as 1/r^2.
• Torque on dipole in external field: tau = p cross E. Dipole aligns with the field at equilibrium.
• Potential energy: U = -p dot E. Minimum (most stable) when p and E are parallel.
• In dielectrics, aligned dipoles produce polarization P (dipole moment per unit volume), which modifies the effective permittivity.

Quick Revision

• p = Qd (C.m), vector from -Q to +Q.
• V = p*cos(theta) / (4*pi*epsilon_0*r^2). Valid for r >> d.
• E_r = 2p*cos(theta) / (4*pi*epsilon_0*r^3), E_theta = p*sin(theta) / (4*pi*epsilon_0*r^3).
• Torque = p cross E (aligns dipole with field). Energy = -p dot E.
• r dependence: monopole V ~ 1/r, dipole V ~ 1/r^2, quadrupole V ~ 1/r^3.
• Exam trap: Potential is maximum along the dipole axis (theta = 0) and zero in the equatorial plane (theta = 90). Field is not zero in the equatorial plane.