The equivalent resistance of the series combination of $ R $ ohm and $ 5 $ ohm is $ 7 $ ohm. Here, $ R $ is __

Question

The equivalent resistance of the series combination of $ R $ ohm and $ 5 $ ohm is $ 7 $ ohm. Here, $ R $ is ________.

Examoogle · Accepted Answer

The correct answer is Option D: 2 ohm. When two or more resistors are connected in series, the equivalent resistance ($R_{eq}$) is equal to the sum of individual resistances. According to the problem, the series combination of $R$ ohm and $5$ ohm gives an equivalent resistance of $7$ ohm. Therefore, $R + 5 = 7$, which gives $R = 2 	ext{ ohm}$.

Step-by-Step Solution & Concepts:
When two or more resistors are connected in series, the equivalent resistance ($R_{eq}$) is equal to the sum of individual resistances. According to the problem, the series combination of $R$ ohm and $5$ ohm gives an equivalent resistance of $7$ ohm. Therefore, $R + 5 = 7$, which gives $R = 2 	ext{ ohm}$.

Working Principle: In a series electrical circuit, the same current flows through each resistor. The total opposition to the flow of charge (equivalent resistance) is the algebraic sum of the individual resistances: $R_{eq} = R_1 + R_2$. Thus, adding resistors in series always increases the total resistance.

Key Formulas: $R_{eq} = R_1 + R_2$ — Equivalent resistance in series combination; $V = I 	imes R_{eq}$ — Ohm's law for the series circuit

Key Points: In a series combination, the equivalent resistance is always greater than any individual resistance.; The current flowing through all resistors connected in series is the same.; The total voltage across the combination is the sum of the potential drops across each resistor.

Advantages: Simple to calculate and connect.; Allows a single switch to control multiple components.

Disadvantages: If one component fails (open circuit), the entire circuit breaks.; Current cannot be controlled individually for each component.

Applications: Old style decorative fairy lights (series strings); Voltage dividers in electronic circuits

Additional Info: Option A (5 ohm): Incorrect, as 5 + 5 = 10 ohm.. Option B (1 ohm): Incorrect, as 1 + 5 = 6 ohm.. Option C (7 ohm): Incorrect, as 7 + 5 = 12 ohm.. Option D (2 ohm): Correct, since 2 + 5 = 7 ohm.

Answer: D is correct — The value of R is 2 ohm since the equivalent resistance in series is the sum of individual resistances ($R + 5 = 7 \implies R = 2$).

Exam Tip: Remember that for parallel combinations, the reciprocal of equivalent resistance is the sum of reciprocals ($\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$), whereas for series, it is a direct algebraic sum.

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