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 defined as: \[ B = \mu_0 \frac{N}{L} I \] where: - \( B \) represents the magnetic field. - \( \mu_0 \) is the permeability of free space. - \( N \) is the total number of turns. - \( L \) is the solenoid's length. - \( I \) is the current through the solenoid. Let's examine how changes in the solenoid's radius and length impact the magnetic field. (i) Impact of Increasing the Radius of the Turns: The formula for the magnetic field does not explicitly include the radius of the solenoid's turns. The magnetic field is determined by the number of turns per unit length and the current, not the radius of individual turns. - Effect: Increasing the radius of the solenoid's turns has no effect on the magnetic field within the solenoid, provided the number of turns per unit length and the current remain constant. (ii) Impact of Doubling the Length and Total Number of Turns: Consider an initial solenoid with length \( L \) and \( N \) turns. If the length is doubled to \( 2L \) and the total number of turns is also doubled to \( 2N \): \[ B_{\text{new}} = \mu_0 \frac{2N}{2L} I = \mu_0 \frac{N}{L} I = B_{\text{initial}} \] - Effect: Doubling both the length and the total number of turns results in no change to the magnetic field within the solenoid, as the ratio \( N/L \) remains constant. Therefore, the magnetic field is unchanged in both scenarios. Final Answer: The magnetic field inside the solenoid is unaffected when: 1. The radius of the solenoid's turns is increased. 2. The length of the solenoid and the total number of turns are both doubled.