Animated Solution for Physics - Magnetic Effects of Current: A perfectly diamagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. The whole system is placed in a uniform magnetic field $\mathbf{B}$. Then, the field inside the paramagnetic substance is

The Ultimate Shield: Understanding Perfect Diamagnetism

Have you ever wondered how we can completely block out a magnetic field? In this problem, we explore a fascinating phenomenon that does exactly that. We are given a perfectly diamagnetic sphere with a small cavity at its center, filled with a paramagnetic substance. The entire setup is placed in a uniform magnetic field, and we need to determine the field experienced by the paramagnetic substance inside the cavity.

Let's break down the components of our system. We have an outer sphere made of a perfectly diamagnetic material. Inside this sphere, right at the geometric center, lies a small cavity. This cavity is filled with a paramagnetic substance.

Normally, a paramagnetic material tends to attract magnetic field lines, slightly enhancing the local magnetic field. However, the crucial part of this problem is the outer shell. The entire system is immersed in a uniform external magnetic field, denoted by $\mathbf{B}$. The question asks us to find the magnetic field inside the cavity.

To solve this, we must understand what it means for a material to be "perfectly diamagnetic." All diamagnetic materials create an induced magnetic field in a direction opposite to an externally applied magnetic field, causing a repulsive effect.

For a perfectly diamagnetic material, this effect is absolute. The magnetic susceptibility, $\chi$, of a perfectly diamagnetic material is exactly $-1$.

The relative permeability, ${\mu }_r$, is related to susceptibility by the equation:

Substituting $\chi =-1$, we get:

Because the relative permeability is zero, the magnetic field inside the material, $B_{in}$, which is given by $B_{in}={\mu }_r{B}_{ext}$, becomes exactly zero.

This complete expulsion of magnetic field lines from the interior of a material is a hallmark of perfect diamagnetism, most famously observed in superconductors as the Meissner effect.

When the external magnetic field $\mathbf{B}$ is applied, the field lines approach the sphere but cannot penetrate it. Instead, they are forced to bend and flow around the outer surface of the sphere. The interior of the diamagnetic sphere remains completely free of any magnetic field.

Since the magnetic field cannot enter the diamagnetic sphere, it certainly cannot reach the cavity located at its center. The outer shell acts as an impenetrable magnetic shield.

Therefore, despite the presence of a paramagnetic substance in the cavity—which would normally interact with a magnetic field—it is completely isolated. The magnetic field inside the paramagnetic substance is absolutely zero.

This elegant concept of magnetic shielding is not just a theoretical curiosity; it is practically applied in designing shields for highly sensitive electronic and magnetic instruments, protecting them from external magnetic interference.