Question

How will the magnetic field inside a long solenoid be affected when:
(i) the radius of the turns of the solenoid is increased,
(ii) the length of the solenoid as well as the total number of its turns are doubled?

Hint:

The magnetic field inside a solenoid is determined by the number of turns per unit length and the current, not the radius of the turns or the total number of turns alone.

Answer 1

The magnetic field \( B \) inside a long solenoid is given by the formula: \[ B = \mu_0 \frac{N}{L} I \] where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the total number of turns,
- \( L \) is the length of the solenoid,
- \( I \) is the current flowing through the solenoid. Now, let’s analyze how changes in the radius and length of the solenoid affect the magnetic field. (i) Effect of Increasing the Radius of the Turns: The radius of the turns of the solenoid does not appear explicitly in the formula for the magnetic field. The magnetic field inside the solenoid depends on the number of turns per unit length and the current, but not on the radius of the individual turns. - Effect: Increasing the radius of the turns of the solenoid does not affect the magnetic field inside the solenoid, as long as the number of turns per unit length and the current remain constant. (ii) Effect of Doubling the Length and the Total Number of Turns: Let the initial length of the solenoid be \( L \) and the initial number of turns be \( N \). Now, we double the length and the total number of turns, i.e., the new length is \( 2L \) and the new number of turns is \( 2N \). Substituting these values into the magnetic field formula: \[ B_{\text{new}} = \mu_0 \frac{2N}{2L} I = \mu_0 \frac{N}{L} I = B_{\text{initial}} \] - Effect: Doubling the length and the total number of turns does not affect the magnetic field inside the solenoid, as the ratio of \( N/L \) remains the same. Thus, the magnetic field remains unchanged in both cases. Final Answer: The magnetic field inside the solenoid remains unchanged when: 1. The radius of the turns of the solenoid is increased. 2. The length of the solenoid and the total number of turns are doubled.