Question

The Insulation Resistance of a 3 km long cable is \(200 \, \text{M}\Omega\). For length of \(15 \, \text{km}\), the Insulation Resistance will be :

• A. \(40 \, \text{M}\Omega\)
• B. \(200 \, \text{M}\Omega\)
• C. \(1000 \, \text{M}\Omega\)
• D. \(50 \, \text{M}\Omega\)

Hint:

For underground cables: \[ R \propto \frac{1}{L} \] Longer cable length means lower insulation resistance.

Answer 1

The Correct Option is A

Concept: Insulation resistance of a cable is inversely proportional to its length. Mathematically: \[ R \propto \frac{1}{L} \] This means:
• if cable length increases,
• insulation resistance decreases. Reason:
• longer cable provides larger leakage path,
• leakage current increases,
• insulation resistance reduces.

Step 1: Writing the proportionality relation. Since: \[ R \propto \frac{1}{L} \] therefore: \[ R_1L_1 = R_2L_2 \]

Step 2: Substituting the given values. Given: \[ R_1 = 200\,\text{M}\Omega \] \[ L_1 = 3\,\text{km} \] \[ L_2 = 15\,\text{km} \] We need to find: \[ R_2 \] Using: \[ R_1L_1 = R_2L_2 \] Substituting: \[ 200 \times 3 = R_2 \times 15 \]

Step 3: Solving for insulation resistance. \[ 600 = 15R_2 \] \[ R_2 = \frac{600}{15} \] \[ R_2 = 40\,\text{M}\Omega \]

Step 4: Interpreting the result physically. The cable length increases from: \[ 3\,\text{km} \rightarrow 15\,\text{km} \] which is: \[ 5 \text{ times} \] Hence insulation resistance becomes: \[ \frac{1}{5} \] of original value: \[ \frac{200}{5}=40\,\text{M}\Omega \]

Step 5: Selecting the correct answer. Therefore: \[ \boxed{40\,\text{M}\Omega} \] Hence correct option is: \[ \boxed{(1)} \]