Question

The percentage increase in magnetic field $ B $ when space within a current-carrying solenoid is filled with magnesium (magnetic susceptibility $ \chi_{mg} = 1.2 \times 10^{-5} $) is:

• A. \( \frac{6}{5} \times 10^{-3} \% \)
• B. \( \frac{5}{6} \times 10^{-5} \% \)
• C. \( \frac{5}{6} \times 10^{-4} \% \)
• D. \( \frac{5}{3} \times 10^{-5} \% \)

Hint:

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

Answer 1

The Correct Option is A

The magnetic field \( B \) in a solenoid is given by the formula: \[ B = \mu_0 n I, \] where: \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length of the solenoid, \( I \) is the current in the solenoid.
Step 1: Magnetic field with material inserted.
When a material with magnetic susceptibility \( \chi_{mg} \) is inserted into the solenoid, the magnetic field in the solenoid increases. The new magnetic field \( B' \) is given by: \[ B' = B (1 + \chi_{mg}), \] where: \( B' \) is the new magnetic field strength after the material is inserted, \( B \) is the magnetic field without the material, \( \chi_{mg} \) is the magnetic susceptibility of the material.
Step 2: Percentage increase in the magnetic field.
The percentage increase in the magnetic field can be calculated as the ratio of the increase in the magnetic field to the original magnetic field, multiplied by 100: \[ \text{Percentage increase} = \left( \frac{B' - B}{B} \right) \times 100 = \chi_{mg} \times 100. \]
Step 3: Substituting the value of \( \chi_{mg} \).
We are given that the magnetic susceptibility \( \chi_{mg} = 1.2 \times 10^{-5} \). Substituting this value into the formula: \[ \text{Percentage increase} = 1.2 \times 10^{-5} \times 100 = 1.2 \times 10^{-3} %. \]
Step 4: Expressing the result.
The percentage increase in the magnetic field is: \[ \frac{6{5} \times 10^{-3} % }. \]
Thus, the correct answer is option (1).

Answer 2

The percentage change in \(B\) is given by: \[ \% \text{ change in } B = \frac{B_{\text{new}} - B_{\text{old}}}{B_{\text{old}}} \times 100\% \] Substitute the values: \[ = \frac{\mu_{\text{ni}} - \mu_{\text{ni0}}}{\mu_{\text{ni0}}} \times 100\% = \frac{\mu - \mu_0}{\mu_0} \times 100\% \] \[ = \frac{(\mu_0 \mu_r - \mu_0)}{\mu_0} \times 100\% \] \[ = (\mu_r - 1) \times 100\% \] Thus, the percentage change is: \[ \chi_n \times 100\% = 1.2 \times 10^{-3} \, \% \] \[ \boxed{\text{Percentage change in } B = 1.2 \times 10^{-3} \, \% } \]