Multipole expansions, calculating the various moments of point charges

• milkism
In summary, a multipole expansion is a mathematical technique used to calculate the electric potential and electric field of a system of point charges by expressing them as an infinite series of terms. The moments of point charges can be calculated using a formula that takes into account the charge and distance of each point charge. The different types of multipole moments include monopole, dipole, quadrupole, octupole, and higher order moments, which represent various aspects of the overall charge distribution. Multipole expansions are used in various areas of physics, particularly in calculating the potential and field of complex charge distributions. They can also be applied to continuous charge distributions by using an integral instead of a summation in the formula.
milkism
Homework Statement
Calculate the mono-di and quadrupole moments of the three charges.
Relevant Equations
See solution.
Problem:

Solution:

This was quite simple, are my solutions correct?

vanhees71, hutchphd and PhDeezNutz
Looks very good to me.

PhDeezNutz and milkism
I love your explanations and motivations. Very thorough.

milkism

1. What is a multipole expansion?

A multipole expansion is a mathematical technique used to describe the behavior of a system of point charges. It involves calculating the various moments of the point charges, which can then be used to approximate the electric potential and field of the system at different distances.

2. How is a multipole expansion calculated?

A multipole expansion is calculated by finding the various moments of the point charges in the system. These moments include the monopole moment (total charge), dipole moment (charge separation), quadrupole moment (charge distribution), and higher order moments. These moments are then used in a series expansion to approximate the electric potential and field.

3. What is the significance of the different moments in a multipole expansion?

The different moments in a multipole expansion represent different aspects of the system of point charges. The monopole moment represents the total charge of the system, the dipole moment represents the charge separation, and the higher order moments represent the charge distribution. These moments are used to approximate the electric potential and field at different distances from the system.

4. How accurate is a multipole expansion?

The accuracy of a multipole expansion depends on the number of moments included in the expansion. The more moments that are included, the more accurate the approximation will be. However, for a system with a large number of point charges, it may be necessary to include a large number of moments to achieve a high level of accuracy.

5. What are some applications of multipole expansions?

Multipole expansions are commonly used in electrostatics, where they can be used to approximate the electric potential and field of a system of point charges. They are also used in quantum mechanics to describe the behavior of atoms and molecules, as well as in astrophysics to model the gravitational potential of celestial bodies.

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