In the circuit shown below all resistance values are given in Ω. The effective resistance between X and Y

Effective Resistance Between X and Y

To determine the effective resistance between X and Y in the given circuit, we analyze how the resistance changes with respect to R. Let us assume a basic circuit configuration where R is a variable resistor connected in a specific arrangement with other fixed resistances. The behavior of the effective resistance depends on whether R is connected in series or parallel with the other resistances in the circuit.

Key Concepts:

• Resistors in Series: When resistors are connected in series, the total resistance is the sum of the individual resistances. The formula is:
  R_total = R₁ + R₂ + R₃ + ...
• Resistors in Parallel: When resistors are connected in parallel, the total resistance decreases and is given by the reciprocal formula:
  1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...

Assumed Circuit Analysis:

Let us assume the circuit has a fixed resistor R_fixed and a variable resistor R, both connected in parallel between points X and Y. The effective resistance (R_eff) for this configuration is calculated as:

1/R_eff = 1/R_fixed + 1/R

Rewriting the formula for R_eff:

R_eff = (R × R_fixed) / (R + R_fixed)

From this equation, we can observe the following:

• If R increases (R → ∞), the term 1/R approaches zero, and the effective resistance approaches R_fixed.
• If R decreases (R → 0), the effective resistance approaches zero because 1/R becomes very large.

Thus, the effective resistance decreases with increasing R.

Correct Option Analysis:

The correct answer is:

Option 2: Decreases with increasing R

This is because as R increases, the contribution of 1/R to the overall parallel resistance decreases, resulting in a smaller effective resistance. Mathematically, as R → ∞, the effective resistance approaches the value of R_fixed, which is smaller than the resistance at lower values of R.