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 defined by \( B = \mu_0 n I \), where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current.
Step 1: Magnetic field with inserted material.
Upon inserting a material with magnetic susceptibility \( \chi_{mg} \), the magnetic field intensifies. The new field, \( B' \), is calculated as \( B' = B (1 + \chi_{mg}) \). Here, \( B' \) represents the enhanced magnetic field strength, \( B \) is the field strength prior to material insertion, and \( \chi_{mg} \) is the material's magnetic susceptibility.
Step 2: Calculating the percentage increase in the magnetic field.
The percentage increase is determined by the ratio of the field's increment to its original value, multiplied by 100: \[ \text{Percentage increase} = \left( \frac{B' - B}{B} \right) \times 100 = \chi_{mg} \times 100. \]
Step 3: Substituting the magnetic susceptibility value.
Given \( \chi_{mg} = 1.2 \times 10^{-5} \), we substitute this into the formula: \[ \text{Percentage increase} = 1.2 \times 10^{-5} \times 100 = 1.2 \times 10^{-3} %. \]
Step 4: Stating the final result.
The magnetic field exhibits a percentage increase of: \[ \frac{6{5} \times 10^{-3} % }. \]
Consequently, option (1) is the correct answer.