Animated Solution for Physics - Magnetic Effects of Current: An iron rod of volume $10^{-3}{\text{ m}}^3$ and relative permeability $1000$ is placed as core in a solenoid with $10\text{ turns/cm}$. If a current of $0.5\text{ A}$ is passed through the solenoid, then the magnetic moment of the rod will be

Imagine you are looking at a simple solenoid—a coil of wire carrying a current. On its own, it generates a modest magnetic field. But the moment you slide an iron rod into its core, something magical happens. The iron rod doesn't just sit there; it becomes highly magnetized, amplifying the magnetic effect tremendously. This problem asks us to find exactly how much magnetic moment this iron core develops.

Iron Inside a Coil

We are given an iron rod with a volume of $V=10^{-3}{\text{ m}}^3$ and a relative permeability of ${\mu }_r=1000$. It is placed inside a solenoid that has a turn density of $n=10\text{ turns/cm}$ and carries a current of $i=0.5\text{ A}$.

When current flows through the solenoid, it creates a magnetic intensity $H$ inside the core. This magnetic intensity forces the magnetic domains within the iron rod to align, giving the rod a net magnetization $I$.

Magnetization and Volume

The total magnetic moment $M$ of any magnetized material is simply the product of its intensity of magnetization $I$ and its volume $V$:

But what is $I$? For a magnetic material, the magnetization is proportional to the applied magnetic intensity $H$:

where $\chi$ is the magnetic susceptibility. We know that ${\mu }_r=1+\chi$. For a highly permeable material like iron where ${\mu }_r=1000$, we can safely approximate $\chi \approx {\mu }_r$. Therefore:

The magnetic intensity $H$ produced by a long solenoid is given by the product of its turn density and the current:

Combining these pieces, we get our master equation for the magnetic moment:

Centimeters to Meters

Before we rush into plugging in the numbers, there is a classic trap waiting for us. The turn density $n$ is given as $10\text{ turns/cm}$. If we use this directly with our volume which is in cubic meters, our units will clash, leading to a completely wrong answer.

We must convert the turn density to standard SI units (turns per meter). Since there are 100 centimeters in a meter, we multiply by 100:

Now, the path is clear. Let's substitute all our values into the master equation:

Notice the beautiful cancellation here. The $1000$ (which is $10^3$) and the $10^{-3}$ from the volume perfectly annihilate each other:

To match the given options, we can write this in scientific notation:

This perfectly matches option (b). By carefully navigating the physics of magnetization and avoiding the unit conversion trap, we've successfully unlocked the magnetic moment of the iron core!