Animated Solution for Physics - Magnetic Effects of Current: Relative permittivity and permeability of a material are ${\epsilon }_r$ and ${\mu }_r$, respectively. Which of the following values of these quantities are allowed for a diamagnetic material?

When we study electromagnetism, we often encounter two fundamental properties that define how a material interacts with electric and magnetic fields: relative permittivity (${\epsilon }_r$) and relative permeability (${\mu }_r$).

Think of relative permittivity as a measure of how much a material resists the formation of an electric field within it. On the other hand, relative permeability tells us how a material responds to an external magnetic field. In this problem, we are tasked with finding the allowed values of these two properties for a specific type of material: a diamagnetic material.

Let's start with relative permittivity, ${\epsilon }_r$. By definition, it is the ratio of the permittivity of a given medium ($\epsilon$) to the permittivity of free space (${\epsilon }_0$).

Free space, or a perfect vacuum, offers the absolute minimum possible resistance to electric field lines. Therefore, no physical material can have a permittivity lower than that of free space. Mathematically, this means $\epsilon \ge {\epsilon }_0$.

When we divide both sides by ${\epsilon }_0$, we get our first golden rule:

This simple inequality is incredibly powerful. It immediately tells us that any value of ${\epsilon }_r$ less than 1 is physically impossible for any material.

Now, let's turn our attention to relative permeability, ${\mu }_r$. How does a diamagnetic material behave in a magnetic field?

Unlike ferromagnetic materials (like iron) that strongly attract magnetic field lines, diamagnetic materials weakly repel them. When placed in an external magnetic field, the electrons in a diamagnetic material slightly adjust their orbits, creating a tiny induced magnetic field that opposes the applied field.

This behavior is quantified by the magnetic susceptibility (${\chi }_m$). For diamagnetic materials, ${\chi }_m$ is a small, negative value ($-1<{\chi }_m<0$). Since relative permeability is related to susceptibility by the equation ${\mu }_r=1+{\chi }_m$, we can deduce our second golden rule:

Armed with our two golden rules, we can now evaluate the given options like a detective eliminating suspects.

- Option (a): ${\epsilon }_r=0.5$ and ${\mu }_r=1.5$. The value of ${\epsilon }_r$ is less than 1, which violates our first rule. This option is impossible. - Option (c): ${\epsilon }_r=0.5$ and ${\mu }_r=0.5$. Again, ${\epsilon }_r$ is less than 1. Impossible. - Option (d): ${\epsilon }_r=1.5$ and ${\mu }_r=1.5$. While ${\epsilon }_r$ is valid, the value of ${\mu }_r$ is greater than 1. This would indicate a paramagnetic or ferromagnetic material, not a diamagnetic one.

This leaves us with Option (b): ${\epsilon }_r=1.5$ and ${\mu }_r=0.5$. Here, ${\epsilon }_r$ is greater than 1, and ${\mu }_r$ perfectly falls in the range between 0 and 1. Both conditions are beautifully satisfied!