Question

A long solenoid with 1000 turns/m has a core material with relative permeability 500 and volume \(10^3\) 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 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.

Hint:

Magnetic moment induced in a material is proportional to its susceptibility \(\chi\), not its permeability \(\mu_r\). For large values, \(\mu_r \approx \chi\), but in precise problems, always use \((\mu_r - 1)\).

Answer 1

Correct Answer: 250

Step 1: Understanding the Concept:
The magnetic moment of a core inside a solenoid is \( M = \chi_m H V \), where \( \chi_m = \mu_r - 1 \) is the magnetic susceptibility, \( H = nI \) is the magnetic field intensity, and \( V \) is the volume.
Step 2: Key Formula or Approach:
Fractional change in magnetic moment = \( \frac{M_2 - M_1}{M_1} \).
Step 3: Detailed Explanation:
Since \( H, n, I, \) and \( V \) are constant, the magnetic moment is proportional to susceptibility: \( M \propto (\mu_r - 1) \).
Initial susceptibility \( \chi_1 = 500 - 1 = 499 \).
Final susceptibility \( \chi_2 = 750 - 1 = 749 \).
Fractional change:
\[ \frac{\Delta M}{M_1} = \frac{\chi_2 - \chi_1}{\chi_1} = \frac{749 - 499}{499} = \frac{250}{499} \]
Given this is in the form \( \frac{x}{499} \), we find \( x = 250 \).
Step 4: Final Answer:
The value of x is 250.