A multipole expansion is a mathematical series used to describe a function that depends on angles—often related to the potential field of a charge distribution—by separating it into components with distinct symmetries. This technique is particularly useful in electromagnetism for approximating the potential at points far from the source by considering the contributions of monopole, dipole, quadrupole, and higher-order moments.
The monopole moment of a point charge distribution is simply the total charge. For a set of point charges, it is calculated by summing the individual charges. Mathematically, if you have point charges \( q_i \) located at positions \( \mathbf{r}_i \), the monopole moment \( Q \) is given by \( Q = \sum_i q_i \).
The dipole moment is a vector quantity that measures the separation of positive and negative charges in a system. For a set of point charges, the dipole moment \( \mathbf{p} \) is calculated as \( \mathbf{p} = \sum_i q_i \mathbf{r}_i \), where \( q_i \) is the charge and \( \mathbf{r}_i \) is the position vector of the \( i \)-th charge. This gives a measure of the overall polarity of the charge distribution.
The quadrupole moment is a rank-2 tensor that provides a more detailed description of the charge distribution's shape than the dipole moment. For point charges, the quadrupole moment tensor \( Q_{ij} \) is calculated as \( Q_{ij} = \sum_i q_i (3 r_{i,j} r_{i,k} - \delta_{jk} r_i^2) \), where \( r_{i,j} \) and \( r_{i,k} \) are the components of the position vector \( \mathbf{r}_i \) of charge \( q_i \), and \( \delta_{jk} \) is the Kronecker delta. This tensor describes how the charge distribution deviates from spherical symmetry.
Multipole expansions are useful in physics because they allow for the simplification of complex problems involving fields generated by charge distributions. By breaking down the potential into monopole, dipole, quadrupole, and higher-order terms, one can often approximate the field at large distances using only the first few terms, significantly reducing the computational complexity. This technique is widely used in electrostatics, gravitational fields