Animated Solution for Physics - Magnetic Effects of Current: The magnetic susceptibility of a material of a rod is $499$. Permeability in vacuum is $4\pi \times 10^{-7}\text{ H/m}$. Absolute permeability of the material of the rod is

Imagine you are standing in a room full of people. Some people are easily influenced by the mood of the room, while others remain completely indifferent. Materials behave in a very similar way when placed inside a magnetic field!

This "influenceability" is measured by a property called Magnetic Susceptibility (${\chi }_m$). It tells us how eagerly a material aligns its internal atomic magnets with an external magnetic field. In our problem, we are given a rod with a magnetic susceptibility of $499$. This is a large, positive number, which immediately tells us that this material is highly cooperative—it's a ferromagnetic material that loves to be magnetized.

To find the Absolute Permeability ($\mu$) of the material, we need to bridge the gap between how the material behaves and the baseline behavior of empty space. The permeability of free space (vacuum) is a universal constant, denoted as ${\mu }_0=4\pi \times 10^{-7}\text{ H/m}$.

The bridge connecting these concepts is the Relative Permeability (${\mu }_r$). Think of relative permeability as a multiplier. It is defined as:

Why the "$1+"$? Because even if a material has zero susceptibility (like a perfect vacuum), it still allows magnetic field lines to pass through it just as well as empty space does (multiplier of $1$).

The absolute permeability is simply the vacuum's permeability scaled up by this relative multiplier:

Combining these, we get our master equation:

Now, it is just a matter of careful substitution and basic arithmetic. Let's plug in our known values:

First, we resolve the bracket. Adding $1$ to $499$ gives us a beautifully round number:

Next, we multiply the constants. $4$ times $500$ is $2000$:

Finally, we need to express our answer in proper scientific notation to match the options. We can rewrite $2000$ as $2\times 10^3$. When we multiply this by $10^{-7}$, the exponents add up ($3-7=-4$):

And there we have it! A straightforward application of fundamental magnetic properties. Always remember the relationship between susceptibility and permeability, as it is a high-yield concept for competitive exams.