Question

The space within the current carrying toroid is filled with aluminium of susceptibility \( \chi \). The percentage increase in the magnetic field \( B \) will be

• A. \( \frac{3}{2} \times 100 \)
• B. \( 2\chi \times 100 \)
• C. \( (1 + \chi) \times 100 \)
• D. \( \chi \times 100 \)

Hint:

When a material with magnetic susceptibility \( \chi \) is inserted into a toroidal coil, the magnetic field increases by a factor of \( (1 + \chi) \), and the percentage increase is \( \chi \times 100 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Magnetic field inside a toroid with vacuum is \( B_0 = \mu_0 n I \). When filled with a material, the field becomes \( B = \mu n I \).
Step 2: Key Formula or Approach:
Relative permeability \( \mu_r = \frac{\mu}{\mu_0} = 1 + \chi \).
Magnetic field in medium \( B = \mu_r B_0 = (1 + \chi) B_0 \).
Step 3: Detailed Explanation:
Increase in magnetic field \( \Delta B = B - B_0 \).
\[ \Delta B = (1 + \chi) B_0 - B_0 = \chi B_0 \]
Percentage increase \( % \Delta B = \frac{\Delta B}{B_0} \times 100 \):
\[ % \Delta B = \frac{\chi B_0}{B_0} \times 100 = \chi \times 100 \]
Step 4: Final Answer:
The percentage increase is \( \chi \times 100 \).