Question

A solenoid of one meter length and \( 3.55 \, \text{cm} \) inner diameter carries a current of \( 5 \, A \). If the solenoid consists of five closely packed layers each with \( 700 \) turns along its length, then the magnetic field at its centre is

• A. 22 mT
• B. 35 mT
• C. 44 mT
• D. 15 mT

Hint:

For calculating the magnetic field inside a long solenoid, the formula \( B = \mu_0 n I \) is fundamental. Ensure that the number of turns per unit length (\( n \)) is correctly determined by dividing the total number of turns by the length of the solenoid. The diameter or radius of the solenoid is generally not required for the magnetic field calculation at the center of a long} solenoid, as long as the length is significantly greater than the diameter. Remember to use the standard value for the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} T m/A} \)) and to convert the final answer to the requested units (e.g., millitesla).

Answer 1

The Correct Option is A

Step 1: Identify the given parameters.
Length of the solenoid, \( L = 1 \, \text{m} \)
Inner diameter = \( 3.55 \, \text{cm} \) (This information about diameter is typically not needed for the magnetic field at the center of a long solenoid, as long as the length is much greater than the diameter, which it is here).
Current, \( I = 5 \, A \)
Number of layers = \( 5 \)
Number of turns per layer = \( 700 \) turns/layer
Step 2: Calculate the total number of turns (N).
Total number of turns, \( N = \text{Number of layers} \times \text{Number of turns per layer} \)
\( N = 5 \times 700 = 3500 \, \text{turns} \).
Step 3: Calculate the number of turns per unit length (n).
The number of turns per unit length, \( n = \frac{\text{Total number of turns}}{\text{Length of the solenoid}} \)
\( n = \frac{N}{L} = \frac{3500 \, \text{turns}}{1 \, \text{m}} = 3500 \, \text{turns/m} \).
Step 4: Use the formula for the magnetic field at the center of a long solenoid.
The magnetic field (\( B \)) at the center of a long solenoid is given by the formula:
\( B = \mu_0 n I \)
where \( \mu_0 \) is the permeability of free space, \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \).
Step 5: Substitute the values and calculate \( B \).
\( B = (4\pi \times 10^{-7} \, \text{T m/A}) \times (3500 \, \text{turns/m}) \times (5 \, \text{A}) \)
\( B = 4 \times 3.14 \times 10^{-7} \times 3500 \times 5 \)
\( B = 4 \times 3.14 \times 10^{-7} \times 17500 \)
\( B = 4 \times 3.14 \times 1.75 \times 10^{-7} \times 10^4 \)
\( B = 4 \times 3.14 \times 1.75 \times 10^{-3} \)
\( B = 12.56 \times 1.75 \times 10^{-3} \)
\( B = 21.98 \times 10^{-3} \, \text{T} \)
\( B \approx 22 \times 10^{-3} \, \text{T} \)
\( B = 22 \, \text{mT} \) (millitesla).