Option d is the answer because when the number of turns is doubled, the inductance will remain the same .so the inductance of the coil increases 4 times.

Explanation:
To understand why the self-inductance quadruples when the number of turns in a solenoid is doubled, we need to consider the concept of self-inductance and its relationship with the number of turns.

Self-Inductance:
Self-inductance, denoted by L, is a property of a circuit that opposes any change in the current flowing through it. It is a measure of the amount of magnetic flux generated by the current in the circuit. The unit of self-inductance is henry (H).

Inductance of a Solenoid:
A solenoid is essentially a coil of wire wound in the form of a cylinder. It generates a magnetic field when a current flows through it. The inductance of a solenoid depends on various factors, including the number of turns, the length of the solenoid, and the cross-sectional area of the solenoid.

Relationship between Self-Inductance and Number of Turns:
The self-inductance of a solenoid is directly proportional to the square of the number of turns. Mathematically, we can represent this relationship as:
L ∝ N^2
Where L is the self-inductance and N is the number of turns.

Doubling the Number of Turns:
Now, let's consider what happens when we double the number of turns in a solenoid. If the original number of turns is N, then the new number of turns will be 2N.

Effect on Self-Inductance:
Using the relationship between self-inductance and the number of turns, we can determine the effect on self-inductance when the number of turns is doubled. Substituting the new number of turns (2N) into the equation, we get:
L ∝ (2N)^2
L ∝ 4N^2

Conclusion:
From the above equation, we can see that the self-inductance quadruples when the number of turns is doubled. Therefore, the correct answer is option 'D' - quadruple.