What is the Formula for Torque on an Electric Dipole in an Electric Field?

The formula for torque (𝜏) on an electric dipole in a uniform electric field is given by the cross product of the electric dipole moment (p) and the electric field strength (E):

𝜏 = p × E

This fundamental equation describes how an electric field exerts a rotational force on an electric dipole, tending to align it with the field.

Understanding the Torque Formula

Torque, a vector quantity, represents the twisting or turning effect of a force. In the context of an electric field, it acts on an electric dipole, which consists of two equal and opposite charges separated by a small distance. The formula 𝜏 = p × E is a vector cross product, meaning both the magnitude and direction of the torque are important.

Let's break down each component of the formula:

𝜏 (Tau): This symbol represents the torque. Torque is a vector quantity that causes an object to rotate. Its SI unit is the Newton-meter (N·m).
p: This symbol represents the electric dipole moment. It is a vector quantity defined as the product of the magnitude of one of the charges (q) and the separation distance (d) between them, directed from the negative charge to the positive charge (p = qd). Its SI unit is the Coulomb-meter (C·m).
E: This symbol represents the electric field strength. It is a vector quantity that describes the force experienced by a unit positive charge at any given point in the field. Its SI unit is Newtons per Coulomb (N/C) or Volts per meter (V/m).

What is an Electric Dipole?

An electric dipole is formed when two equal but opposite electric charges are separated by a small, fixed distance. For instance, a positive charge (+q) and a negative charge (-q) placed a distance 'd' apart constitute an electric dipole. Many molecules, like water (H₂O), are inherently electric dipoles duea to the uneven distribution of charges within them. The dipole moment (p) quantifies this separation and magnitude of charges.

The Cross Product: Magnitude and Direction

The cross product in the formula 𝜏 = p × E provides crucial information about both the strength and the orientation of the torque.

Magnitude of Torque

The magnitude of the torque (𝜏) can be calculated using the following scalar equation:

𝜏 = pE sin(θ)

Where:

p is the magnitude of the electric dipole moment.
E is the magnitude of the electric field strength.
θ (theta) is the angle between the electric dipole moment vector (p) and the electric field vector (E).
Maximum Torque: Torque is maximum when sin(θ) is 1, which occurs when θ = 90° (i.e., when the dipole moment is perpendicular to the electric field).
Zero Torque: Torque is zero when sin(θ) is 0, which occurs when θ = 0° or θ = 180° (i.e., when the dipole moment is aligned or anti-aligned with the electric field). In these orientations, the dipole is in equilibrium.

Direction of Torque

The direction of the torque vector (𝜏) is perpendicular to the plane containing both the electric dipole moment vector (p) and the electric field vector (E). It can be determined using the right-hand rule:

Point the fingers of your right hand in the direction of the first vector (p).
Curl your fingers towards the direction of the second vector (E).
Your thumb will point in the direction of the resulting torque vector (𝜏).

This torque acts to rotate the electric dipole, tending to align the dipole moment vector (p) with the electric field vector (E).

Practical Applications and Significance

Understanding torque on electric dipoles is critical in various fields:

Molecular Physics: It explains the behavior of polar molecules in external electric fields, influencing their alignment and interaction.
Dielectric Materials: The alignment of molecular dipoles in a dielectric material under an electric field is fundamental to how capacitors work and how dielectrics enhance electric field effects.
Spectroscopy: Analyzing how molecules rotate in response to electric fields can provide insights into their structure and properties.

Summary of Variables

Symbol	Quantity	SI Unit	Description
𝜏	Torque	Newton-meter (N·m)	Rotational force that tends to align the dipole with the electric field.
p	Electric Dipole Moment	Coulomb-meter (C·m)	Vector representing the separation of charge in an electric dipole.
E	Electric Field Strength	N/C or V/m	Strength and direction of the electric field.
θ	Angle	Radians or Degrees	Angle between p and E.