Question

An electric kettle has two heating coils. When one of the coils is connected to an A.C source, the water in the kettle boils in $10min$. When the other coil is used the water boils in $40min$. The amount of heat produced by the individual coil or the parallel combination is the same. If both the coils are connected in parallel, the time taken for the same quantity of water to boil will be

Answer 1

Step 1: Given data

1. Time is taken by the kettle to boil water when the first coil is active is $t_1=10min.$
2. Time is taken by the kettle to boil water when the second coil is active is $t_2=40min.$
3. In this case, voltage and heat production is constant.

Step 2: Joule's law of heating

1. Joule's law states that when a current I is flowing in a conductor of resistance R for a time T sec, then the produced heat on the conductor is directly proportional to the square of the electric current.
2. Joule's law is defined by the form, $H=\frac{V^2}{R}T$ , where, H is the produced heat on the conductor.

Step 3: Diagram

Step 4: Calculation of resistance

Let $R_1$ and $R_2$ be the two resistance of the kettle. Now from Joule's law,

$H=\frac{V^2}{R}T$

For the first coil of resistance,

$H=\frac{V^2}{R_1}\times 10....................\left(1\right)$

For the second coil of resistance,

$H=\frac{V^2}{R_2}\times 40....................\left(2\right)$

Equating (1) and (2) we get,

$$\begin{gathered} \frac{10}{R_1}=\frac{40}{R_2} \\ or{R}_2=4R_1..............\left(3\right) \end{gathered}$$

Step 5: Calculation of Heat

We know in parallel combination, the equivalence resistance is $R=\frac{R_1{R}_2}{R_1+R_2}$

When both are connected in parallel, then heat will be produced,

$$\begin{gathered} H=\frac{V^2}{{\displaystyle \frac{R_1{R}_2}{R_1+R_2}}}\times t \\ orH=\frac{V^2\times 5R_1}{4{R_1}^2}\times t \\ orH=\frac{V^2\times 5}{4R_1}\times t.............\left(4\right) \end{gathered}$$ (using equation 3 )

Step 6: Calculation of time

Comparing equation (4) with equation (2) we get,

$$\begin{gathered} 10=\frac{5t}{4} \\ ort=\frac{10}{5}\times 4=8 \\ ort=8min. \end{gathered}$$

Therefore, the time taken to boil the water in the kettle is $8min.$