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 magnetic field inside a long solenoid is uniform and given by \(B = \mu_0 n I\), where \(n\) is the number of turns per unit length.

Step 2: Detailed Explanation:
Given: Current, \(I = 1 A\). Length of solenoid, \(L = 1 m\). Total number of turns, \(N = 7000\). Number of turns per unit length, \(n = \frac{N}{L} = \frac{7000}{1} = 7000 \, \text{turns/m}\). Permeability of free space, \(\mu_0 = 4\pi \times 10^{-7} \, Tm/A\). Magnetic field, \(B = \mu_0 n I = (4\pi \times 10^{-7}) \times 7000 \times 1\). \[ B = 4 \times 3.14 \times 10^{-7} \times 7000 = 4 \times 3.14 \times 7 \times 10^{-4} = 4 \times 21.98 \times 10^{-4} \approx 87.92 \times 10^{-4} = 8.792 \times 10^{-3} T \] Approximately \(8.8 \times 10^{-3} T\).

Step 3: Final Answer:
The magnetic field is \(8.8 \times 10^{-3} T\).