1. Concept Overview
An electric dipole placed in an electric field feels a twisting effect called torque. Even though the net force on the dipole can be zero in a uniform electric field, the two charges experience opposite forces that create a tendency to rotate. This torque tries to align the dipole with the electric field direction.
2. Why Does a Dipole Experience Torque?
In a uniform electric field, the positive charge feels a force in the direction of the field, and the negative charge feels a force in the opposite direction. These two equal and opposite forces act at different points, creating a twisting effect.
3. Definition of Torque on a Dipole
Torque on a dipole in a uniform electric field is given by the product of dipole moment, electric field strength, and the sine of the angle between them.
\( \tau = pE \sin \theta \)
Here, \( \theta \) is the angle between the dipole moment vector \( \vec{p} \) and the electric field \( \vec{E} \).
4. Vector Form of Torque
The torque can also be expressed using the cross product:
\( \vec{\tau} = \vec{p} \times \vec{E} \)
This shows that torque is perpendicular to the plane formed by \( \vec{p} \) and \( \vec{E} \).
5. Conditions for Equilibrium
Depending on how the dipole is oriented in the electric field, torque may try to rotate it or keep it steady.
5.1. Stable Equilibrium
Occurs when \( \theta = 0^\circ \), meaning the dipole moment is aligned with the field.
Torque is zero because \( \sin 0 = 0 \), and any slight rotation brings the dipole back to alignment.
5.2. Unstable Equilibrium
Occurs when \( \theta = 180^\circ \), meaning the dipole is opposite to the field direction.
Torque is also zero, but even a small disturbance causes rotation toward the stable position.
6. Potential Energy of a Dipole in an Electric Field
The potential energy depends on how the dipole is oriented in the electric field.
6.1. Formula
\( U = -pE \cos \theta \)
The energy is minimum when the dipole is aligned with the field and maximum when it is opposite.
7. Angle Dependence of Torque
The value of torque changes with angle:
7.1. Special Cases
- \( \theta = 0^\circ \): \( \tau = 0 \)
- \( \theta = 90^\circ \): \( \tau = pE \) (maximum)
- \( \theta = 180^\circ \): \( \tau = 0 \)
8. Worked Examples
8.1. Example 1: Calculating Torque
A dipole has moment \( p = 5 \times 10^{-8} \text{ C·m} \). It is placed in an electric field \( E = 3000 \text{ N/C} \) at an angle of \( 45^\circ \).
Torque:
\( \tau = pE \sin \theta = 5 \times 10^{-8} \times 3000 \times \sin 45^\circ \)
\( \tau = 1.06 \times 10^{-4} \text{ N·m} \)
8.2. Example 2: Maximum Torque
If the dipole moment is \( 4 \times 10^{-8} \text{ C·m} \) and \( E = 500 \text{ N/C} \), maximum torque occurs at \( 90^\circ \):
\( \tau_{\max} = pE = 4 \times 10^{-8} \times 500 = 2 \times 10^{-5} \text{ N·m} \).
9. Physical Interpretation
Torque tries to rotate the dipole so that its positive end points in the direction of the electric field. This behaviour is similar to how a compass needle aligns with Earth’s magnetic field, except here the alignment is due to electric forces. Understanding torque helps explain how molecules orient in fields and why certain materials respond to electric fields the way they do.