Question

A long solenoid of radius 1 mm has 100 turns per mm. If 1 A current flows in the solenoid, the magnetic field strength at the centre of the solenoid is:

• A. 6.28 x 10^-2 T
• B. 12.56 x 10^-2 T
• C. 12.26 x 10^-4 T
• D. 6.28 x 10^-4 T

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A solenoid is a long coil of wire wrapped around a cylinder.
For an ideal or "long" solenoid, the magnetic field inside is nearly uniform and directed along the axis.
The strength of the magnetic field at the center of the solenoid depends only on the permeability of free space, the number of turns per unit length, and the current magnitude.
Note that the radius of the solenoid does not affect the magnetic field strength at its center, provided it is a long solenoid.
Key Formula or Approach:
The magnetic field (\(B\)) inside a long solenoid is given by Ampere's Circuital Law as:
\[ B = \mu_0 n I \]
Where:
\( \mu_0 \) is the permeability of free space (\(4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}\)).
\( n \) is the number of turns per unit length (turns per meter).
\( I \) is the current flowing through the solenoid.
Step 2: Detailed Explanation:
First, we must express all given quantities in SI units.
1. Calculate the turns per unit length (\(n\)):
The problem gives 100 turns per mm.
Since \(1 \text{ m} = 1000 \text{ mm}\), the number of turns per meter is:
\[ n = 100 \frac{\text{turns}}{\text{mm}} \times \frac{1000 \text{ mm}}{1 \text{ m}} = 100,000 \text{ turns/m} = 10^5 \text{ turns/m} \]
2. Identify the current (\(I\)):
The current \(I\) is given as \(1 \text{ A}\).
3. Substitute values into the formula:
\[ B = (4\pi \times 10^{-7}) \times (10^5) \times (1) \]
Combine the powers of 10:
\[ B = 4\pi \times 10^{-7+5} = 4\pi \times 10^{-2} \text{ T} \]
4. Numerical Calculation:
Taking \(\pi \approx 3.14\):
\[ B = 4 \times 3.14 \times 10^{-2} \text{ T} \]
\[ B = 12.56 \times 10^{-2} \text{ T} \]
Step 3: Final Answer:
The magnetic field strength at the center of the solenoid is \(12.56 \times 10^{-2} \text{ T}\).