# Moment (physics)

In physics, a moment is an expression involving the product of a distance and another physical quantity, and in this way it accounts for how the physical quantity is located or arranged.

Moments are usually defined with respect to a fixed reference point; they deal with physical quantities as measured at some distance from that reference point. For example, the moment of force acting on an object, often called torque, is the product of the force and the distance to the object (i.e., the reference point). In principle, any physical quantity can be multiplied by distance to produce a moment; commonly used quantities include forces, masses, and electric charge distributions.

## Elaboration

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

${\displaystyle \mu _{n}=r^{n}\,Q,}$

where ${\displaystyle Q}$ 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's density over space:

${\displaystyle \mu _{n}=\int r^{n}\,\rho (r)\,dr}$

where ${\displaystyle \rho }$ 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 ${\displaystyle r^{n}\,\rho (r)}$ 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 ${\displaystyle r}$ is measured, although for certain moments (technically, the lowest non-zero moment) this dependence vanishes 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 first moment: ${\displaystyle \mathbf {\tau } =rF}$, or, more generally, ${\displaystyle \mathbf {r} \times \mathbf {F} }$
• Similarly, angular momentum is the 1st moment of momentum: ${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }$. Note that momentum itself is not a moment.
• The electric dipole moment is also a 1st moment: ${\displaystyle \mathbf {p} =q\,\mathbf {d} }$ for two opposite point charges or ${\displaystyle \int \mathbf {r} \,\rho (\mathbf {r} )\,d^{3}r}$ for a distributed charge with charge density ${\displaystyle \rho (\mathbf {r} )}$

Moments of mass:

• The total mass is the zeroth moment of mass
• The center of mass is the 1st moment of mass normalized by total mass: ${\displaystyle \mathbf {R} ={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}}$ for a collection of point masses, or ${\displaystyle {\frac {1}{M}}\int \mathbf {r} \rho (\mathbf {r} )\,d^{3}r}$ for an object with mass distribution ${\displaystyle \rho (\mathbf {r} )}$
• The moment of inertia is the 2nd moment of mass: ${\displaystyle I=r^{2}m}$ for a point mass, ${\displaystyle \sum _{i}r_{i}^{2}m_{i}}$ for a collection of point masses, or ${\displaystyle \int r^{2}\rho (\mathbf {r} )\,d^{3}r}$ for an object with mass distribution ${\displaystyle \rho (\mathbf {r} )}$. Note that the center of mass is often (but not always) taken as the reference point.

## 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:

${\displaystyle \Phi (\mathbf {r} )=\int {\frac {\rho (\mathbf {r'} )}{|\mathbf {r} -\mathbf {r'} |}}\,d^{3}r'=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)q_{\ell m}\,{\frac {Y_{\ell m}(\theta ,\varphi )}{r^{\ell +1}}}}$

The coefficients ${\displaystyle q_{\ell m}}$ are known as multipole moments, and take the form:

${\displaystyle q_{\ell m}=\int (r')^{\ell }\,\rho (\mathbf {r'} )\,Y_{\ell m}^{*}(\theta ',\varphi ')\,d^{3}r'}$

where ${\displaystyle \mathbf {r} '}$ expressed in spherical coordinates ${\displaystyle (r',\varphi ',\theta ')}$ 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.)

When ${\displaystyle \rho }$ represents an electric charge density, the ${\displaystyle q_{lm}}$ are, in a sense, projections of the moments of electric charge: ${\displaystyle q_{00}}$ is the monopole moment; the ${\displaystyle q_{1m}}$ are projections of the dipole moment, the ${\displaystyle q_{2m}}$ are projections of the quadrupole moment, etc.

## Applications of multipole moments

The multipole expansion applies to 1/r scalar potentials, examples of which include the electric potential and the gravitational potential. For these potentials, the expression can be used to approximate the strength of a field produced by a localized distribution of charges (or mass) by calculating the first few moments. For sufficiently large r, a reasonable approximation can be obtained from just the monopole and dipole moments. Higher fidelity can be achieved by calculating higher order moments. Extensions of the technique can be used to calculate interaction energies and intermolecular forces.

The technique can also be used to determine the properties of an unknown distribution ${\displaystyle \rho }$. 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,[2][3] but has also been applied to the universe itself,[4] 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.[5][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. However, historical evolution of the term 'moment' and its use in different branches of science, such as mathematics, physics and engineering, is unclear.

Federico Commandino, in 1565, translated into Latin from Archimedes:

The center of gravity of each solid figure is that point within it, about which on all sides parts of equal moment stand.[6]

This was apparently the first use of the word moment (Latin, momentorum) in the sense which we now know it: a moment about a center of rotation.[7]

The word moment was first used in Mechanics in its now rather old-fashioned sense of 'importance' or 'consequence,' and the moment of a force about an axis meant the importance of the force with respect to its power to generate in matter rotation about the axis... But the word 'moment' has also come to be used by analogy in a purely technical sense, in such expressions as the 'moment of a mass about an axis,' or 'the moment of an area with respect to a plane,' which require definition in each case. In those instances there is not always any corresponding physical idea, and such phrases stand, both historically and scientifically, on a different footing. – A. M. Worthington, 1920[8]