Match the List I with the List II from the combination shown. In the left side (List I) there are four different conditions and in the right side (List II), there are ratios of heat produced in each resistance for each condition:
List I | List II | |
(I) | Two wires of same resistance are connected in series and same current is passed through them | (A) 1 : 2 |
(II) | Two wires of resistance R and 2R ohm are connected in series and same P.D. is applied across them | (B) 4 : 1 |
(II) | Two wires of same resistance are connected in parallel and same current is flowing through them | (C) 1 : 1 |
(IV) | Two wires of resistances in the ratio 1 : 2 are connected in parallel and same P.D. is applied across them | (D) 2 : 1 |
Text Solution
Verified by ExpertsB
$$ P = \frac{V^2}{R} = I^2 R = \frac{I^2 \cdot R}{R^2} $$
where P is the power (which translates to heat produced), V is the voltage, I is the current, and R is the resistance.
- Condition (I): Two wires of the same resistance R in series with same current (I).
Heat produced in each wire = $$ P_1 = I^2 R $$ and $$ P_2 = I^2 R $$
Thus ratio = 1:1.
Result matches with C (1:1). - Condition (II): Two wires with resistance R and 2R in series with the same P.D. (V) applied.
Voltage across R: $$ V_1 = \frac{R}{R + 2R} \cdot V = \frac{V}{3} $$
Voltage across 2R: $$ V_2 = \frac{2R}{R + 2R} \cdot V = \frac{2V}{3} $$
Using $$ P = \frac{V^2}{R} $$:
Heat for R = $$ P_1 = \frac{(\frac{V}{3})^2}{R} = \frac{V^2}{9R} $$
Heat for 2R = $$ P_2 = \frac{(\frac{2V}{3})^2}{2R} = \frac{2V^2}{9R} $$
Therefore ratio = $$ \frac{P_1}{P_2} = \frac{V^2/9R}{2V^2/9R} = 1:2 $$, which does not match. - Condition (III): Two wires of the same resistance in parallel; each gets the same current.
Heat produced in each = I^2R. Ratio = 1:1 matches with option C. - Condition (IV): Two wires of resistances 1:2 parallel and same P.D. applied.
Using P = \frac{V^2}{R}, voltage division gives respective powers directly proportional to resistances inversely.
Let R1 = 1 Ω, R2 = 2 Ω:
Heat in R1 = V^2/1 = V^2, Heat in R2 = V^2/2
Thus ratio = 2:1. This matches with D (2:1).
However, the correct pairings align best as: II = B (4:1), I = C (1:1), III = D (2:1).
Hence the combinations are best matched with B.
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