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. What is the ratio of heat produced in series and parallel combinations?

Answer 1

Step 1: Given parameters

Two conducting wires, are made of the same material and have equal lengths and diameters.

That means they have the same resistance $R$.

Let $R_s$ is equivalent resistance when they are connected in series and $R_p$ is equivalent resistance when they are connected in parallel.

Step 2: Calculate the resistance and power in series connection.

For two resistance in series Combination:

$\begin{array}{rcl}{R}_s& =& R_1+R_2\\ & =& R+R\\ & =& 2R\end{array}$

Now the power is given by

$\begin{array}{rcl}{P}_{series}& =& \frac{V^2}{R_s}\\ & =& \frac{V^2}{2R}.........\left(1\right)\end{array}$

Step 3: Calculate the resistance and power in parallel connection.

For two resistance in parallel Combination:

$\begin{array}{rcl}\frac{1}{R_p}& =& \frac{1}{R_1}+\frac{1}{R_2}\\ & =& \frac{1}{R}+\frac{1}{R}\\ & =& \frac{2}{R}\end{array}$

So we can write, $R_p=\frac{R}{2}$.

Now the power is given by

$\begin{array}{rcl}{P}_{parallel}& =& \frac{V^2}{R_P}\\ & =& \frac{V^2}{{\displaystyle \frac{R}{2}}}\\ & =& \frac{2V^2}{R}.......\left(2\right)\end{array}$

Step 4: Find the ratio of the heat produced in series and the parallel combination.

Now the ratio of heat produced (power consumed in 1 second) in series and parallel combinations can be given by dividing, (2) from (1), i.e.

$\begin{array}{rcl}\frac{P_{series}}{P_{parallel}}& =& \frac{{\displaystyle \frac{V^2}{2R}}}{{\displaystyle \frac{2V^2}{R}}}\\ & =& \frac{1}{4}\end{array}$

Hence, the ratio of heat produced in series and the parallel combination is $\frac{1}{4}$.