Question

The insulation resistance of a single core cable is \(160\text{ M}\Omega/\text{km}\). The insulation resistance for \(4\text{ km}\) length is

• A. \(640\text{ M}\Omega\)
• B. \(160\text{ M}\Omega\)
• C. \(40\text{ M}\Omega\)
• D. \(80\text{ M}\Omega\)

Hint:

Always remember: Longer cable = Lower insulation resistance. This is because a longer cable provides more parallel paths for leakage current to escape to the ground.

Answer 1

The Correct Option is C

Concept: Unlike the conductor resistance of a cable, which increases linearly with length, the insulation resistance ($R_{ins}$) of a cable behaves inversely with respect to its total length ($l$). This inverse relationship occurs because increasing the length of the cable expands the cross-sectional surface area available for leakage currents to flow radially from the core conductor to the outer sheath. The insulation resistance is given by the formula: \[ R_{ins} = \frac{\rho}{2\pi l} \ln\left(\frac{R_2}{R_1}\right) \] From this formula, we can establish that: \[ R_{ins} \propto \frac{1}{l} \implies R_1 l_1 = R_2 l_2 \]

Step 1: Identify the parameters provided in the question.
* Initial base insulation resistance per unit length (\(R_1\)) = \(160\text{ M}\Omega\) * Initial reference line length (\(l_1\)) = \(1\text{ km}\) * Target operational length (\(l_2\)) = \(4\text{ km}\)

Step 2: Apply the inverse proportionality relationship to calculate the new resistance.
\[ R_2 = R_1 \times \left(\frac{l_1}{l_2}\right) \] Substituting the known values into this equation: \[ R_2 = 160\text{ M}\Omega \times \left(\frac{1\text{ km}}{4\text{ km}}\right) \] \[ R_2 = \frac{160}{4} = 40\text{ M}\Omega \] Therefore, when the cable length increases to \(4\text{ km}\), the total insulation resistance drops to \(40\text{ M}\Omega\). This matches Option (C).