Reviewed by: JJ Ben-Joseph

Compute Missing Quantity

The concept of an electric dipole plays a central role in electrostatics, molecular physics, and materials science. A dipole consists of two equal and opposite charges separated by a small distance. The product of charge magnitude and separation defines the dipole moment $p$, a vector pointing from negative to positive charge. When such a dipole is placed in a uniform electric field $E$, the field exerts forces on the charges that create a net torque attempting to align the dipole with the field lines. The magnitude of this torque is given by $\tau =pE\sin \theta$, where $\theta$ is the angle between the dipole moment and the field direction. This calculator evaluates the relationship in either direction so you can determine any missing variable.

Dipoles appear in many physical systems. In chemistry, the separation of charge within a molecule gives rise to a permanent dipole moment, influencing how the molecule interacts with external fields and neighboring molecules. For example, water has a substantial dipole moment that underlies its ability to dissolve ionic compounds. In solid-state physics, dipoles contribute to dielectric polarization, affecting the capacitance of materials and the behavior of ferroelectric devices. Understanding the torque experienced by a dipole in a field helps explain phenomena such as the orientation of polar molecules in liquids and the operation of electric motors that exploit dielectric materials.

The torque equation can be derived from the fundamental definition of torque $\tau =r\times F$. Each charge in the dipole experiences a force $F=qE$ in opposite directions. The lever arm for each force is half the separation distance $d$. Combining these results yields $\tau =pE\sin \theta$ with $p=qd$. Although our calculator uses this compact form, recognizing its origin reinforces the connection between electric forces and rotational dynamics.

Using the calculator is straightforward. Enter numerical values for any three of the four fields: dipole moment, electric field, angle, or torque. Leave the unknown quantity blank and press the compute button. The script verifies that exactly one field is empty. Angles should be provided in degrees, and the internal code converts them to radians for computation. If you supply a torque and wish to solve for the angle, the program uses the inverse sine function; because the sine function is periodic, multiple angles can produce the same torque, so the returned angle is the principal value between 0° and 180°.

Units deserve careful attention. Dipole moment is measured in coulomb-meters, a product of charge and distance. Small molecular dipoles are often expressed in debye, where $1 D=3.33564\times {10}^{-}C·m$. Electric field strength is in newtons per coulomb, equivalent to volts per meter. Torque is in newton-meters, the standard SI unit for rotational effects. The calculator does not perform unit conversions automatically, so ensure all inputs are in compatible SI units before computing.

Consider a simple example: a dipole moment of $2\times {10}^{-}C·m$ (roughly comparable to a small polar molecule) placed in a uniform electric field of 5 kV/m. If the dipole initially makes a 60° angle with the field, the torque is approximately $4.33\times {10}^{-}$ N·m. Such tiny torques are nonetheless significant at molecular scales, where thermal agitation competes with field-induced alignment. By comparing the calculated torque to thermal energies, one can estimate whether a given field is sufficient to orient molecules in a solution.

Torque on an electric dipole is closely related to potential energy. The potential energy of a dipole in a field is $U=-pE\cos \theta$. As the dipole rotates to align with the field, its potential energy decreases, and the lost energy converts to kinetic energy or dissipates as heat. While our calculator focuses on torque, understanding the energy landscape helps explain equilibrium orientations and the behavior of dipoles in varying fields. At equilibrium, the torque is zero because $\theta =\mathrm{0°}$ or $\mathrm{180°}$, corresponding to stable and unstable alignments respectively.

In macroscopic systems, arrays of dipoles can produce measurable mechanical effects. Dielectric slabs in capacitor plates experience torques that can cause them to rotate if not securely mounted. Electrostatic actuators exploit dipole interactions to produce motion in microelectromechanical systems. In biology, many proteins have permanent dipole moments that interact with electric fields in cellular environments, affecting folding and orientation near membranes. The simple relation handled by this calculator thus spans scales from nanoscopic chemistry to large-scale electromechanical devices.

When using this tool for educational purposes, it is instructive to vary the angle while keeping dipole moment and field fixed. Observe how the torque follows a sine curve, peaking at 90° and vanishing at 0° and 180°. This behavior mirrors that of a compass needle in a magnetic field, providing a tangible analogy. You can also explore how increasing the field or dipole moment scales the torque linearly, reinforcing the proportional nature of the interaction.

The table below presents sample calculations for different dipole moments and field strengths at a fixed angle of 45°. These examples illustrate the order of magnitude of torques encountered in molecular and macroscopic situations.

p (C·m)	E (kV/m)	θ (°)	τ (N·m)
2×10⁻²⁹	5	45	7.07×10⁻²⁶
1×10⁻¹⁸	1	45	7.07×10⁻¹⁹
1×10⁻⁶	0.1	45	7.07×10⁻⁸

Ultimately, the torque formula encapsulates how electric fields influence rotational dynamics. Whether you are studying molecular orientation, designing sensors, or analyzing dielectric materials, the ability to compute any variable in $\tau =pE\sin \theta$ provides quick insight. This calculator performs the math instantly in your browser, preserving privacy and functioning without an internet connection.