Two resistors of 20 ohm and 30 ohm are connected in parallel across the same potential difference. What is the effective resistance of the combination?

Introduction / Context:
Combining resistors in series and parallel is a key skill in circuit analysis. Many real circuits use combinations rather than single resistors, so being able to calculate the effective or equivalent resistance helps determine current and voltage distributions. This question asks for the effective resistance of two resistors, 20 ohm and 30 ohm, connected in parallel. It tests your knowledge of the parallel resistance formula and your ability to apply it correctly to a numerical example.

Given Data / Assumptions:

• Two resistors have resistances R1 = 20 ohm and R2 = 30 ohm.
• They are connected in parallel across the same potential difference.
• We need to find the effective resistance R_eq of the parallel combination.
• We assume ideal resistors with no additional circuit elements.

Concept / Approach:
For resistors connected in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances. The formula is 1 / R_eq = 1 / R1 + 1 / R2. After calculating this sum, we invert it to obtain R_eq. An important property is that the equivalent resistance of resistors in parallel is always less than the smallest individual resistance in the group. Using these principles, we can compute the effective resistance for 20 ohm and 30 ohm connected in parallel.

Step-by-Step Solution:
Step 1: Write the parallel resistance formula for two resistors: 1 / R_eq = 1 / R1 + 1 / R2.Step 2: Substitute R1 = 20 ohm and R2 = 30 ohm into the formula.Step 3: Compute 1 / R_eq = 1 / 20 + 1 / 30.Step 4: Find a common denominator, which is 60: 1 / 20 = 3 / 60 and 1 / 30 = 2 / 60, so 1 / R_eq = 3 / 60 + 2 / 60 = 5 / 60.Step 5: Invert the fraction to get R_eq = 60 / 5 = 12 ohm.

Verification / Alternative check:
You can verify that the answer makes sense by checking the property that the equivalent resistance of a parallel combination is less than the smallest individual resistor. Here, the smallest resistor is 20 ohm, and our result is 12 ohm, which is indeed smaller. Another way to verify is to consider an applied voltage, for example 60 V. The current through the 20 ohm resistor would be 3 A, and through the 30 ohm resistor 2 A, giving a total current of 5 A. The equivalent resistance would then be R_eq = V / I_total = 60 / 5 = 12 ohm, confirming the calculation.

Why Other Options Are Wrong:
Option A, 50 ohm, is simply the sum of the resistances and would be the correct answer for series connection, not parallel. Option C, 24 ohm, might result from incorrectly averaging the two resistances, which is not the correct method. Option D, 36 ohm, does not follow from any standard formula and is larger than the smaller resistor, which cannot be correct for a parallel combination. Only option B, 12 ohm, matches the result obtained from the proper parallel resistance formula.

Common Pitfalls:
A frequent error is to add the resistances directly for parallel combinations, confusing them with series combinations. Another mistake is to compute a simple arithmetic mean instead of using the reciprocal formula. To avoid these errors, remember the key rule: in series, resistances add directly; in parallel, their reciprocals add. Also, use the useful check that the equivalent resistance in parallel must be less than the smallest individual resistance. This provides a quick sanity check on your numerical answer.

Final Answer:
12 ohm