Question

A solenoid of length 50 cm and radius 10 cm has two closely wound layers of windings 100 turns each. If a current of 2.5 A is passing through the windings, the magnetic field (in \( 10^{-4} \) T) at a point 5 cm from the axis is

• A. \( 2\pi \)
• B. 31.4
• C. \( 4\pi \)
• D. Zero

Hint:

The field inside a solenoid is independent of the distance from the axis as long as the point is well inside. Add the turn densities of all layers.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The magnetic field inside a long solenoid is uniform and parallel to the axis. The point 5 cm from the axis is inside the solenoid (since radius is 10 cm). The total magnetic field is due to both layers of windings.
Step 2: Key Formula or Approach:
Magnetic field inside a solenoid: \( B = \mu_o n I \). Where \( n = \frac{N}{L} \) is turns per unit length. For multiple layers, \( N \) is the total number of turns or simply add the fields if layers are in series. Here, we use total turns density.
Step 3: Detailed Explanation:
Given: Length \( L = 50 \, \text{cm} = 0.5 \, \text{m} \). Total turns \( N = 100 + 100 = 200 \) (two layers of 100 each). Current \( I = 2.5 \, \text{A} \). Permeability \( \mu_o = 4\pi \times 10^{-7} \, \text{Tm/A} \). Calculate \( n \): \[ n = \frac{N}{L} = \frac{200}{0.5} = 400 \, \text{turns/m} \] Calculate \( B \): \[ B = \mu_o n I \] \[ B = (4\pi \times 10^{-7}) \times 400 \times 2.5 \] \[ B = 4\pi \times 10^{-7} \times 1000 \] \[ B = 4\pi \times 10^{-4} \, \text{T} \] The question asks for the value in \( 10^{-4} \, \text{T} \). Value \( = 4\pi \).
Step 4: Final Answer:
The magnetic field is \( 4\pi \times 10^{-4} \) T.