Moment (physics)

Not to be confused with Momentum.
For the mathematical concept, see Moment (mathematics).

In physics, moment is a tendency to produce motion, especially, a rotation about a point or axis. Moments are generally defined with respect to a fixed reference point; they deal with physically measurable quantities such as forces or electric charge distributions in relation to that reference point.

Elaboration

In its most simple and basic form, a moment is the product of a distance from some reference point, raised to some power, multiplied by some physical quantity such as a force, charge distribution, etc.:

$\mu_n = r^n\,Q$,

where $Q$ is the physical quantity such as a force applied at a point, or a point charge, or a point mass, etc. If the quantity is not concentrated solely at a single point, the moment is the integral of that quantity over space:

$\mu_n=\int r^n\,\rho(r)\,dr$

where $\rho$ is the distribution of the density of charge, mass, or whatever quantity is being considered.

More complex forms take into account the angular relationships between the distance and the physical quantity, but the above equations capture the essential feature of a moment, namely the existence of an underlying $r^n\,\rho(r)$ or equivalent term. This implies that there are multiple moments (one for each value of n) and that the moment generally depends on the reference point from which the distance $r$ is measured, although for certain moments (technically, the lowest non-zero moment) this dependence cancels and the moment becomes independent of the reference point.

Each value of n corresponds to a different moment: the 1st moment corresponds to n=1; the 2nd moment to n=2, etc. The 0th moment (n=0) is sometimes called the monopole moment; the 1st moment (n=1) is sometimes called the dipole moment, and the 2nd moment (n=2) is sometimes called the quadrupole moment, especially in the context of electric charge distributions.

Examples:

• The moment of force, or torque, is a 1st moment: $\mathbf{\tau} = rF$, or, more generally, $\mathbf{r} \times \mathbf{F}$
• The electric dipole moment is also a 1st moment: $\mathbf{p} = q\,\mathbf{d}$ for two opposite point charges or $\int \mathbf{r}\,\rho(\mathbf{r})\,d^3r$ for a distributed charge with charge density $\rho(\mathbf{r})$
• The moment of inertia is a 2nd moment: $I = r^2 m$ for a point mass, $\sum_i r_i^2 m_i$ for a collection of point masses, or $\int r^2\rho(\mathbf{r}) \, d^3r$ for an object with mass distribution $\rho(\mathbf{r})$

Multipole Moments

Assuming a density function that is finite and localized to a particular region, outside that region a 1/r potential may be expressed as a series of spherical harmonics:

$\Phi(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d^3r' = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left( \frac{4\pi}{2l+1} \right) q_{lm}\, \frac{Y_{lm}(\theta, \phi)}{r^{l+1}}$

The coefficients $q_{lm}$ are known as multipole moments, and take the form:

$q_{lm} = \int (r')^{l}\, \rho(\mathbf{r'})\, Y^*_{lm}(\theta',\phi')\, d^3r'$

where $\mathbf{r}'$ expressed in spherical coordinates $(r',\phi',\theta')$ is a variable of integration. A more complete treatment may be found in pages describing multipole expansion or spherical multipole moments. (Note: the convention in the above equations was taken from Jackson.[1] The conventions used in the referenced pages may be slightly different.)

This expression applies to 1/r potententials, examples of which include the electric potential, the magnetic potential and the gravitational potential. The expression can be used, for example, to approximate the strength of a field produced by a complex distribution of charges by calculating the first few multipole moments. For sufficiently large r, a good approximation can be obtained from just the monopole and dipole moments (for electric and gravitational fields; classical magnetic fields have no monopole moment).

This technique can also be used to determine the properties of an unknown distribution $\rho$. Measurements pertaining to multipole moments may be taken and used to infer properties of the underlying distribution. This technique applies to small objects such as molecules, but has also been applied to the universe itself,[2] being for example the technique employed by the WMAP and Planck experiments to analyze the Cosmic microwave background radiation.

History

The concept of moment in physics is derived from the mathematical concept of moments.[3][clarification needed]. The principle of moments is derived from Archimedes' discovery of the operating principle of the lever. In the lever one applies a force, in his day most often human muscle, to an arm, a beam of some sort. Archimedes noted that the amount of force applied to the object, the moment of force, is defined as M = rF, where F is the applied force, and r is the distance from the applied force to object.[4] However, historical evolution of the term 'moment' and its use in different branches of science, such as mathematics, physics and engineering, is unclear.