Explanation:

Capacitance:
Capacitance is a property of a capacitor that determines its ability to store electrical energy in the form of an electric field. It is defined as the ratio of the electric charge on each conductor to the potential difference between the conductors.

Relation between Capacitance and Length of Cable:
The capacitance of a cable is directly proportional to its length. This means that as the length of the cable increases, the capacitance also increases.

Mathematical Representation:
The capacitance of a cable can be mathematically represented as follows:

C = (ε * A) / d

Where:
C is the capacitance of the cable,
ε is the permittivity of the material between the conductors,
A is the area of the cross-section of the cable,
d is the distance between the conductors.

Effect of Doubling the Length of Cable:
When the length of a cable is doubled, it means that the distance between the conductors is also doubled. Let's assume the initial distance between the conductors is 'd'. After doubling the length, the new distance between the conductors becomes '2d'.

Relation between Capacitance and Distance:
The capacitance of a cable is inversely proportional to the distance between the conductors. This means that as the distance between the conductors increases, the capacitance decreases.

Mathematical Representation:
The capacitance of a cable with a distance 'd' between the conductors can be mathematically represented as follows:

C1 = (ε * A) / d

The capacitance of the cable after doubling the distance '2d' between the conductors can be mathematically represented as follows:

C2 = (ε * A) / (2d)

Comparison of Capacitance:
To compare the capacitance of the cable before and after doubling the length, we can calculate the ratio of C2 to C1.

C2 / C1 = [(ε * A) / (2d)] / [(ε * A) / d]
C2 / C1 = d / (2d)
C2 / C1 = 1/2

Conclusion:
From the above calculation, it can be concluded that when the length of a cable is doubled, the capacitance becomes double. Hence, the correct answer is option 'D' - the capacitance becomes double.