It seems amazing that
Insulation Resistance of a Cable is inversely proportional to the length of
Cable. As per our general perception, the resistance is directly proportional
to length of conductor but mind here that it is not the resistance of conductor
rather it is the resistance of the Insulation provided around the conductor.

We can understand this by
general thinking as, suppose there is a cable of length 100 m and it is having
a insulation resistance of 500 Mohm. Now for this length of cable the leakage
current will flow throughout the cable as shown in figure below. Here we are
talking about leakage current because we are concerned with insulation
resistance.

Now, if we cut the cable
into two equal halves, then the leakage current through the insulation of cable
will reduce by half as there is no full length of cable available for leakage
current. Therefore, we can say that Insulation Resistance has increased two
times i.e. to 1000 Mohm for our example.

Now, we will see this aspect
of insulation resistance mathematically. Consider a single core cable of radius
R1 and internal sheath radius R2. Let L be the length of cable and ρ be the
resistivity of the insulation.

For calculating the
insulation resistance, we take an infinitesimally small layer of insulation of
thickness dx at a radius R from the centre of the cable conductor. The length
of insulation from where leakage current will flow is dx and area of
cross-section will be 2πxL (A cylinder of length L and radius x).

Therefore,

Insulation Resistance of the
assumed layer = ρdx/ 2πxL (From Resistance
= ρL/A)

Now, for getting the
insulation Resistance of whole cable we need to integrate up to a radius of R2
starting from R1 and to a length L.

Insulation Resistance, R

*Thus it is clear from the above expression that Insulation resistance is Inversely Proportional to the length of Cable.*

Thank you!

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