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 combination would be?
A. 1:2
B. 2:1
C. 1:4
D. 4:1

Answer 1

Hint: Resistance R is inversely proportional to the heat produced in the circuit. Find the equivalent resistance of the wires when connected in series and parallel. Apply the formula of electrical power(heat produced).
Formula used:
${{R}_{s}}={{R}_{1}}+{{R}_{2}}+{{R}_{3}}+{{R}_{4}}+.........$
$\dfrac{1}{{{R}_{p}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}+\dfrac{1}{{{R}_{3}}}+\dfrac{1}{{{R}_{4}}}.......$
$P=\dfrac{{{V}^{2}}}{R}$
${{P}_{p}}=\dfrac{{{V}^{2}}}{{{R}_{p}}}$

Complete answer:
As the material, length and diameters of the two wires are the same hence, resistance would also be same in both the wires.
Let the resistance in them be R.
Therefore when connected in series,
${{R}_{s}}=R+R=2R$
Therefore when connected in parallel,
$\dfrac{1}{{{R}_{p}}}=\dfrac{1}{R}+\dfrac{1}{R}=\dfrac{2}{R}$ ,
OR${{R}_{p}}=\dfrac{R}{2}$
Formula for Electrical power is $P=\dfrac{{{V}^{2}}}{R}$
Power or heat produced in series circuit is${{P}_{s}}=\dfrac{{{V}^{2}}}{{{R}_{s}}}$
Power or heat produced in parallel circuit is${{P}_{p}}=\dfrac{{{V}^{2}}}{{{R}_{p}}}$
Therefore the ratio between the heat produced in series and parallel and series circuit is
$\dfrac{{{P}_{p}}}{{{P}_{s}}}=\dfrac{{{V}^{2}}/{{R}_{p}}}{{{V}^{2}}/{{R}_{s}}}=\dfrac{{{R}_{s}}}{{{R}_{p}}}=\dfrac{2R}{R/2}=\dfrac{4}{1}$
${{P}_{s}}:{{P}_{p}}=4:1$ .

So, the correct answer is “Option D”.

Additional Information:
The rate at which electric energy is consumed by the circuit is known as electrical power.
A circuit is called parallel when there are two or more parts through which current can flow.
A series circuit has only one path for flow of current.

Note:
When calculating equivalent resistance for parallel circuits, remember that it is always inverse at first. Resistance is the same as the build of the wire is the same. Electrical power is also a kind of heat production.