Question

If a current of 1 A is passed through a 1 m long solenoid of 7000 turns, the magnetic field produced at the middle of the solenoid is

• A. \(2.2 \times 10^{-3} T\)
• B. \(4.4 \times 10^{-3} T\)
• C. \(7.0 \times 10^{-4} T\)
• D. \(8.8 \times 10^{-3} T\)
• E. \(14.0 \times 10^{-4} T\)

Hint:

For a solenoid, \(B = \mu_0 n I\) is valid for points near the center if the length is much greater than the radius.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for the magnetic field inside a long solenoid. A solenoid is a coil of wire, and when current flows through it, it produces a nearly uniform magnetic field inside, especially near the center.
Step 2: Key Formula or Approach:
The formula for the magnetic field (\(B\)) inside a long solenoid is:
\[ B = \mu_0 n I \] where:
- \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7}\) T·m/A).
- \(n\) is the number of turns per unit length.
- \(I\) is the current flowing through the wire.
The number of turns per unit length (\(n\)) is calculated as \(n = \frac{N}{L}\), where \(N\) is the total number of turns and \(L\) is the length of the solenoid.
Step 3: Detailed Explanation:
First, let's list the given values:
- Current, \(I = 1\) A.
- Length of the solenoid, \(L = 1\) m.
- Total number of turns, \(N = 7000\).
- Permeability of free space, \(\mu_0 = 4\pi \times 10^{-7}\) T·m/A.
Next, calculate the number of turns per unit length (\(n\)):
\[ n = \frac{N}{L} = \frac{7000 \text{ turns}}{1 \text{ m}} = 7000 \text{ turns/m} \] Now, substitute these values into the formula for the magnetic field:
\[ B = \mu_0 n I \] \[ B = (4\pi \times 10^{-7} \, \text{T·m/A}) \times (7000 \, \text{m}^{-1}) \times (1 \, \text{A}) \] \[ B = 4\pi \times 7000 \times 10^{-7} \, \text{T} \] \[ B = 28000\pi \times 10^{-7} \, \text{T} \] \[ B = 2.8\pi \times 10^4 \times 10^{-7} \, \text{T} \] \[ B = 2.8\pi \times 10^{-3} \, \text{T} \] To get a numerical value, use the approximation \(\pi \approx 3.14159\):
\[ B \approx 2.8 \times 3.14159 \times 10^{-3} \, \text{T} \] \[ B \approx 8.796 \times 10^{-3} \, \text{T} \] This value is approximately \(8.8 \times 10^{-3}\) T.
Step 4: Final Answer:
The magnetic field produced at the middle of the solenoid is approximately \(8.8 \times 10^{-3}\) T, which corresponds to option (D).