Problem 32 - Master Resource Book in JEE Main Physics

Question

The self inductance \(L\) of a solenoid of length \(l\) and area of cross-section \(A\), with a fixed number of turns \(N\) increases as (a) \(l\) and \(A\) increase (b) \(l\) decreases and \(A\) increases (c) \(l\) increases and \(A\) decreases (d) both \(I\) and \(A\) decreases

Step-by-Step Solution

Answer

(b) \(l\) decreases and \(A\) increases.

Step 1: Understand the Formula for Self Inductance

The self inductance \( L \) of a solenoid is given by the formula \[ L = \mu_0 \frac{N^2 A}{l} \] where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( A \) is the area of cross-section, and \( l \) is the length of the solenoid.

Step 2: Analyze the Effect of Changing Parameters

From the formula, \( L \) is directly proportional to the area \( A \) and inversely proportional to the length \( l \). This means that increasing \( A \) will increase \( L \), whereas increasing \( l \) will decrease \( L \). Conversely, decreasing \( l \) will increase \( L \), and decreasing \( A \) will decrease \( L \).

Step 3: Identify the Correct Option

Given this relationship, the condition that increases \( L \) is when \( l \) decreases and \( A \) increases. This corresponds to option (b).

Key Concepts

Solenoid

A solenoid is a coil of wire designed to generate a magnetic field. It is generally long and cylindrical, bearing resemblance to a spring, made by winding a series of loops or turns of wire closely spaced around a cylindrical object. The solenoid creates a uniform magnetic field in its interior when an electric current is passed through it.
A solenoid's ability to induce a magnetic field arises from the concept of electromagnetic induction. This principle is widely used in various applications such as electromagnets, transformers, and sensors, showcasing the solenoid's usefulness in both practical and theoretical fields.

• A solenoid works as a controller: It can control door locks, actuators, and other mechanical devices.
• It serves as a primary component in various magnetic field-related experiments, providing a consistent and manageable magnetic field source.

Area of Cross-Section

The area of cross-section of a solenoid is the surface area facing perpendicular to its length at any given point. It is typically circular, given the common cylindrical shape of solenoids. Understanding the area of cross-section is crucial as it directly affects the solenoid's self inductance.
When we increase the area of cross-section, the number of magnetic lines of force that the solenoid can accommodate increases. This results in an increase in the solenoid's inductance. Hence, the self inductance of the solenoid depends directly on the area of its cross-section.

• Self-inductance is represented by the formula \( L = \mu_0 \frac{N^2 A}{l} \), where \( A \) stands for the area of cross-section.
• An increase in cross-sectional area \( A \) will lead to an increase in inductance \( L \).

Length of Solenoid

The length of a solenoid refers to the distance over which the coils are uniformly distributed. It plays a significant role in determining the solenoid’s magnetic field and its inductance. The length is directly correlated with how the magnetic field is distributed and how the inductive properties of the solenoid are affected.
As described in the self-inductance formula, the longer the solenoid, the lesser the inductance when all other factors are constant. This is because the inductance \( L \) is inversely proportional to the length \( l \) of the solenoid.

• Long solenoids produce a more uniform magnetic field inside the coil compared to shorter solenoids.
• When the length \( l \) is reduced, for a fixed cross-sectional area \( A \), the inductance \( L \) increases, leading to a stronger inductive property.