A magnetic field B0, when applied on an atom with a...

A magnetic field $B_{0}$, when applied on an atom with a spherically symmetric charge distribution $\rho(r)$, induces a diamagnetic current. Show that the magnetic field produced by the diamagnetic current at the nucleus is $$ \Delta B_{0}=-\left(\frac{e B_{0}}{3 m}\right) V_{0} $$ where $V_{0}$ is the electrostatic potential at the nucleus, given by $$ V_{0}=\int_{0}^{\infty} \frac{\rho(r)}{r} 4 \pi r^{2} d r $$

A magnetic field $B_{0}$, when applied on an atom with a spherically symmetric charge distribution $\rho(r)$, induces a diamagnetic current. Show that the magnetic field produced by the diamagnetic current at the nucleus is
$$
\Delta B_{0}=-\left(\frac{e B_{0}}{3 m}\right) V_{0}
$$
where $V_{0}$ is the electrostatic potential at the nucleus, given by
$$
V_{0}=\int_{0}^{\infty} \frac{\rho(r)}{r} 4 \pi r^{2} d r
$$

Key Concepts

Diamagnetism

Diamagnetism is a form of magnetism that arises in all materials due to the orbital motion of electrons. When an external magnetic field is applied, these electrons adjust their orbits, creating small currents that generate magnetic fields opposing the external field. This phenomenon is intrinsic to atoms and is a universal response, even though in many materials this effect is very weak compared to other forms of magnetism.

Diamagnetic Current

The diamagnetic current is the induced current that occurs in an atom when a magnetic field is applied. The perturbation caused by the magnetic field alters the motion of the electrons in their orbitals, leading to a systematic current loop that creates its own magnetic field. This induced field, as calculated in the problem, is directly related to the electrostatic potential at the nucleus and acts in opposition to the applied magnetic field.

Electrostatic Potential from Charge Distribution

In the context of atomic physics, the electrostatic potential at the nucleus is derived from the integral of the spherically symmetric charge distribution. It quantifies the overall effect of the electronic cloud’s distribution on the potential experienced at a particular point (here, the nucleus). This potential is essential for determining how the induced diamagnetic current influences the local magnetic field.

Magnetic Field Induced by Currents

Calculating the magnetic field produced by any current distribution involves integrating the contributions of current elements. In atomic systems, this means relating the induced diamagnetic current to the magnetic field at the nucleus through established electromagnetic principles. This concept bridges the microscopic motion of electrons with observable magnetic effects, such as the counteracting field experienced at the nucleus.

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