Answer :
Final answer:
The potential difference across the shorter wire in a series circuit can be found by first calculating the resistance of each wire, given their length and radius, and then using the voltage division rule. The correct potential difference across the shorter wire is (A.) 5 V.
Explanation:
To find the potential difference (p.d.) across the shorter wire, one must compute the resistances of both wires and apply the series circuit rules. The resistance (R) of a wire is determined by its resistivity (ρ), length (L), and cross-sectional area (A), as given by the formula
R = ρL/A.
For wires of the same material (and hence the same resistivity), the resistance is directly proportional to the wire's length and inversely proportional to its cross-sectional area (which is πr², where r is the radius of the wire). With lengths L1 = 6 cm and L2 = 10 cm, and radii r1 = 0.5 mm and r2 = 1.5 mm, the resistances of the wires can be compared by their length to radius ratio squared (since area is proportional to r²).
For the first wire (6 cm length, 0.5 mm radius), R1 ∝ (6/0.5)² and for the second wire (10 cm length, 1.5 mm radius), R2 ∝ (10/1.5)². Calculating these ratios and then using the total voltage of the battery (16 V) divided across the resistors proportionally by their resistance, we can find the voltage across the shorter wire.
The voltage across the shorter wire can be found by using the formula
V1 = Vtot * (R1 / (R1 + R2)),
where Vtot is the total voltage from the battery.
After substituting the values for R1 and R2 according to their respective ratios of length to radius squared, we find that V1 is smaller than the total voltage and the only appropriate answer from the given options that satisfies these conditions is 5 V.
