Question

An electric kettle has two coils, when one of these is switched on the water in the kettle boils in 6 min. When the other coil is switched on, the water boils in 3 min. If the two coils are connected in series, the time taken to boil the water in the kettle is

Answer 1

Step 1: Given data

1. Time is taken by the kettle to boil water when the first coil is active is $t_1=6min.$
2. Time is taken by the kettle to boil water when the second coil is active is $t_2=3min.$
3. In this case voltage and heat production is constant.

Step 2: Formula Used

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.

Diagram

From the figure when the switch is open it is considered a parallel connection because the voltage is divided into two resistor $R_1$ and $R_2$.

when the switch is closed it is considered a series connection because the same voltage is carried by that loop which contain $R_1$ and $R_2$ resistors.

Step 3: Calculation

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,

$H=\frac{V^2}{R_1}\times 6....................\left(1\right)$

For the second coil of resistance,

$H=\frac{V^2}{R_2}\times 3....................\left(2\right)$

Equating (1) and (2) we get,

$$\begin{gathered} \frac{6}{R_1}=\frac{3}{R_2} \\ or{R}_1=2R_2..............\left(3\right) \end{gathered}$$

Calculation of time

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

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

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

$$\begin{gathered} \frac{t}{3}=3 \\ ort=9min. \end{gathered}$$

Therefore, the time taken to boil the water in the kettle is $9min.$