Question

In an experiment, the interior of a first current-carrying toroid with \(1000\) turns per unit length is filled with aluminum (susceptibility \(2.4\times 10^{-5}\)). A second, identical toroid with \(500\) turns per unit length is filled with magnesium (susceptibility \(1.2\times 10^{-5}\)). If both toroids carry the same current \(I\), the ratio of the increase in their magnetic fields is

• A. \(2:1\)
• B. \(4:1\)
• C. \(1:4\)
• D. \(1:2\)

Hint:

The increase in the field is \(\mu_0 n I \chi\), so compare \(n\chi\) for the two toroids.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
With a core of susceptibility \(\chi\), the toroid field is \(B=\mu_0nI(1+\chi)\). Without the core, it is \(\mu_0nI\).

Step 2: Increase in the field
\(\Delta B=\mu_0nI\chi\). For the same current \(I\), \(\Delta B\propto n\chi\).

Step 3: Calculation
\(\dfrac{\Delta B_1}{\Delta B_2}=\dfrac{1000\times2.4\times10^{-5}}{500\times1.2\times10^{-5}}=\dfrac{2\times2}{1}=4\).

Final Answer:
The ratio of the increase in the magnetic fields is 4:1. \[ \boxed{\text{(B) }4:1} \]