Animated Solution for Physics - Magnetic Effects of Current: A long solenoid with $1000\text{ turns/m}$ has a core material with relative permeability $500$ and volume $10^3{\text{ cm}}^3$. If the core material is replaced by another material having relative permeability of $750$ with same volume maintaining same current of $0.75\text{ A}$ in the solenoid, the fractional change in the magnetic moment of the core would be approximately $\left(\frac{x}{499}\right)$. Find the value of $x$.

The problem presents a classic scenario in electromagnetism: a solenoid filled with a magnetic core material. When current flows through the solenoid, it generates a magnetic field, which in turn magnetizes the core. The question asks us to find the fractional change in the magnetic moment of the core when we swap out the material for another one with a different relative permeability, while keeping everything else (current, volume, turn density) constant.

When a magnetic material is placed inside a solenoid, the total magnetic field inside is not just due to the current in the wires. The material itself gets magnetized and contributes to the field.

The external driving field created by the solenoid is called the magnetic intensity ($H$). It depends purely on the geometry of the solenoid and the current: where $n$ is the number of turns per unit length and $i$ is the current. Since the problem states that both the current and the solenoid remain the same, $H$ is constant.

The core material responds to this magnetic intensity by developing its own magnetisation ($I$), which is the magnetic moment per unit volume. The relationship between magnetisation and magnetic intensity is governed by the material's magnetic susceptibility ($\chi$):

Susceptibility is directly related to the relative permeability (${\mu }_r$) given in the problem:

Let's calculate the susceptibility for both materials: For the first material: ${\chi }_1=500-1=499$ For the second material: ${\chi }_2=750-1=749$

The total magnetic moment ($m$) of the core is simply its magnetisation multiplied by its volume ($V$):

Since $H$ and $V$ are constant for both materials, the magnetic moment is directly proportional to the susceptibility:

This is a beautiful simplification! We don't even need to use the given values for volume ($10^3{\text{ cm}}^3$) or current ($0.75\text{ A}$). They are just distractors. We can directly write the ratio of the magnetic moments:

The question asks for the fractional change in the magnetic moment. This is defined as the change in magnetic moment divided by the initial magnetic moment:

Substituting our ratio into this expression:

The problem states that this fractional change is approximately $\frac{x}{499}$. By comparing our result, we can clearly see that: $x=250$

This problem elegantly demonstrates how understanding proportionalities can save you from tedious calculations. By recognizing which variables remain constant, we bypassed the need to calculate the actual magnetic moments and arrived at the answer using simple ratios.