67. The magnetic field in the interior of a long current-carrying solenoid can be increased by

(A) increasing the length of solenoid while keeping the number of turns per unit length as constant

(B) increasing the number of turns per unit length

(C) increasing the current in the solenoid

(D) decreasing the length of solenoid while keeping the number of turns per unit length as constant

Magnetic Field Inside a Long Current-Carrying Solenoid – Complete Theory, Formula

Correct Answer

(B) and (C)

Magnetic Field Inside a Long Solenoid

A solenoid is a long cylindrical coil consisting of many closely wound turns of insulated wire. When electric current flows through the wire, each turn produces a magnetic field. Since the magnetic fields produced by all the turns add together inside the solenoid, the resultant magnetic field becomes strong and almost uniform throughout its interior.

For an ideal long solenoid, the magnetic field inside the solenoid is given by

B = μ₀ n I

where

• B = Magnetic field inside the solenoid
• μ₀ = Permeability of free space
• n = Number of turns per unit length
• I = Current flowing through the solenoid

This equation is derived using Ampère’s Circuital Law and is valid for a sufficiently long solenoid where edge effects can be neglected.

Factors Affecting the Magnetic Field

The equation B = μ₀nI clearly shows that the magnetic field depends only on three quantities:

• The permeability of the medium inside the solenoid.
• The number of turns per unit length (n).
• The current flowing through the solenoid (I).

Notice that the equation does not contain the total length of the solenoid or the total number of turns separately. It depends only on the turn density, that is, the number of turns per unit length.

Detailed Option-Wise Analysis

Option (A): Increasing the Length of the Solenoid While Keeping the Number of Turns per Unit Length Constant

Suppose the length of the solenoid is increased while the number of turns per unit length remains unchanged. This means that as the length increases, the total number of turns also increases proportionally so that the turn density n remains constant.

Since the magnetic field depends only on n and I, increasing the total length alone does not change the magnetic field.

Therefore, the magnetic field inside the solenoid remains exactly the same.

Option (A) is Incorrect.

Option (B): Increasing the Number of Turns per Unit Length

The magnetic field is directly proportional to the number of turns per unit length.

If more turns are wound in the same length, the magnetic fields produced by individual turns overlap more effectively, producing a stronger resultant magnetic field.

Mathematically,

B ∝ n

Hence, increasing the turn density directly increases the magnetic field.

Option (B) is Correct.

Option (C): Increasing the Current in the Solenoid

The magnetic field is also directly proportional to the current flowing through the solenoid.

Every current-carrying turn produces its own magnetic field. As the current increases, the magnetic field produced by each turn becomes stronger. Since all these fields add together, the resultant magnetic field inside the solenoid also increases.

Mathematically,

B ∝ I

Therefore, increasing the current increases the magnetic field proportionally.

Option (C) is Correct.

Option (D): Decreasing the Length While Keeping the Number of Turns per Unit Length Constant

Reducing the length of the solenoid while keeping the turn density constant simply reduces the total number of turns. However, the number of turns per unit length remains unchanged.

Since the magnetic field depends only on n and not on the total length, decreasing the length under this condition does not affect the magnetic field.

Therefore, the magnetic field remains unchanged.

Option (D) is Incorrect.

Physical Interpretation

Inside a long solenoid, each circular turn contributes a magnetic field in the same direction along the axis of the solenoid. Because the turns are closely packed, these magnetic fields reinforce one another, creating a strong and nearly uniform magnetic field throughout the interior.

If the turns are packed more closely (higher turn density), more magnetic field lines pass through each unit length, increasing the overall magnetic field. Similarly, increasing the electric current strengthens the magnetic field produced by every individual turn.

On the other hand, merely changing the total length without altering the turn density does not affect the magnetic field because the number of turns per unit length remains the same.

Important Formula

B = μ₀nI

If a magnetic core having relative permeability μ_r is inserted inside the solenoid, the magnetic field becomes

B = μ₀μ_rnI

This explains why iron-core electromagnets produce much stronger magnetic fields than air-core solenoids.

Exam-Oriented Key Concepts

Students should remember that the magnetic field inside a long solenoid depends directly on the current and the number of turns per unit length. It is independent of the total length of the solenoid as long as the turn density remains constant. Questions based on this concept are very common in competitive examinations because they test conceptual understanding of the solenoid formula rather than simple memorization.

Final Answer

The magnetic field inside a long current-carrying solenoid can be increased by increasing the number of turns per unit length and by increasing the current flowing through the solenoid.

Correct Options: (B) and (C)