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Question
Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be:
Options
1 : 2
2 : 1
1 : 4
4 : 1
Solution
1 : 4
Explanation:
Since both the wires are made of the same material and have equal lengths and equal diameters, they have the same resistance. Let it be R.
When Connected in series, their equivalent resistance is given by
Rs = R + R =2R
When connected in parallel, their equivalent resistance is given by
`1/("R"_p) = 1/"R" + 1/"R"`
= `2/"R"` or `"R"_p = "R"/2`
Further, electrical power is given by,
`"p" = ("V"^2)/"R"`
Power or heat produced in series, `"p"_s = ("V"^2)/("R"_s)`
Power or heat produced in parallel, `"p"_p = ("V"^2)/("R"_p)`
Thus, `("p"_s)/("p"_p) = (("V"^2)/("R"_s))/(("V"^2)/("R"_p))`
= `("R"_p)/("R"_s)`
= `("R"/2)/(2"R")`
= `1/4`
or Ps : Pp = 1 : 4
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