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As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality ...
Position vectors r and r′ used in the calculation. The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: =, = where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and is the D'Alembert operator. [2]
A theoretical magnetic point dipole has a magnetic field of exactly the same form as the electric field of an electric point dipole. A very small current-carrying loop is approximately a magnetic point dipole; the magnetic dipole moment of such a loop is the product of the current flowing in the loop and the (vector) area of the loop.
This earlier time in which an event happens such that a particle at location r 'sees' this event at a later time t is called the retarded time, t r. The retarded time varies with position; for example the retarded time at the Moon is 1.5 seconds before the current time and the retarded time on the Sun is 500 s before the current time on the Earth.
A key point is that the potential of the dipole falls off faster with distance R than that of the point charge. The electric field of the dipole is the negative gradient of the potential, leading to: [ 7 ] E ( R ) = 3 ( p ⋅ R ^ ) R ^ − p 4 π ε 0 R 3 . {\displaystyle \mathbf {E} \left(\mathbf {R} \right)={\frac {3\left(\mathbf {p} \cdot ...
The electric potential at any location, r, in a system of point charges is equal to the sum of the individual electric potentials due to every point charge in the system. This fact simplifies calculations significantly, because addition of potential (scalar) fields is much easier than addition of the electric (vector) fields.
The time-dependent behavior of magnetization becomes important when considering nanoscale and nanosecond timescale magnetization. Rather than simply aligning with an applied field, the individual magnetic moments in a material begin to precess around the applied field and come into alignment through relaxation as energy is transferred into the ...
Before progressing in the math, and trying to find a more explicit expression for the proportionality constant (), there's an important aspect that we need to discuss. That is that we have found that the potential felt by an atom in a light induced potential follows the square of the time averaged electric field.