Question

The value of aluminium susceptibility is $2.2 \times 10^{-5}$. The percentage increase in the magnetic field if space within a current carrying toroid is filled with Aluminium is $\frac{x}{10^4}$. Then the value of x is __________.

Hint:

For paramagnetic materials, the magnetic field increases by a factor of \((1 + \chi)\). The susceptibility \(\chi\) directly represents the relative change in the field.

Answer 1

Correct Answer: 22

Step 1: Understanding the Concept:
The magnetic field inside a toroid in vacuum is \(B_0 = \mu_0 H\).
When filled with a material of susceptibility \(\chi\), the field becomes \(B = \mu H = \mu_0 (1 + \chi) H\).
Step 2: Key Formula or Approach:
Fractional increase in field = \(\frac{B - B_0}{B_0} = \chi\).
Percentage increase = \(\chi \times 100%\).
Step 3: Detailed Explanation:
Given \(\chi = 2.2 \times 10^{-5}\).
The percentage increase is:
\[ \text{Percentage increase} = 2.2 \times 10^{-5} \times 100 = 2.2 \times 10^{-3} \]
We are given that this percentage increase is \(\frac{x}{10^4}\).
Equating the two:
\[ \frac{x}{10^4} = 2.2 \times 10^{-3} \]
\[ x = 2.2 \times 10^{-3} \times 10^4 = 22 \]
Step 4: Final Answer:
The value of \(x\) is 22.