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 $10$ minutes. When the other coil is used the water boils in $40$ minutes. If both the coils are connected in parallel, the time taken by the same quantity of water to boil will be

Question

In a meter bridge experiment, a resistance of $ 10 \, \Omega $ is balanced by a resistance $ X $ with a balance point at $ 40 \, \mathrm{cm} $. What is the value of $ X $?

Answer 1

To solve the problem of determining the time taken by the water to boil when both heating coils are connected in parallel, we need to understand the relationship between power and time. The key is to recognize that the power (and thus heating effect) of the coils can be related to their time to boil the water individually. The concept involves the following steps:

The power required to boil the water is inversely proportional to the time taken, i.e., if a coil takes less time to boil the water, its power is greater.
Let the power of the first coil be P_1 and it takes 10 minutes to boil the water. Therefore, the energy required to boil the water using the first coil is given by: E = P_1 \times 10.
Similarly, let the power of the second coil be P_2. It takes 40 minutes to boil the same quantity of water, so the energy required is: E = P_2 \times 40.
Since both instances are boiling the same amount of water, the energy E is the same in both scenarios. Therefore, equating both expressions for energy, we have: P_1 \times 10 = P_2 \times 40.
From this, we derive the relation: P_1 = 4P_2.
Now consider both coils are connected in parallel. The total power P_{\text{total}} would be the sum of the individual powers: P_{\text{total}} = P_1 + P_2.
Substitute the value of P_1: P_{\text{total}} = 4P_2 + P_2 = 5P_2.
The time t_{\text{parallel}} taken for the water to boil when both coils are used can be found using the relationship that the energy is constant: E = P_{\text{total}} \times t_{\text{parallel}}. Since the energy E is the same for the quantity of water to boil, equate the energy required using one of the single coils: E = P_1 \times 10 = P_{\text{total}} \times t_{\text{parallel}}.
Substitute P_{\text{total}} = 5P_2 and simplify: 4P_2 \times 10 = 5P_2 \times t_{\text{parallel}}. Hence, t_{\text{parallel}} = \frac{4 \times 10}{5} = 8 minutes.

Thus, when both coils are used together in parallel, the water will boil in 8 minutes. This matches option (a) 8 minutes.

Answer 2

To solve the problem of determining the time taken by the water to boil when both heating coils are connected in parallel, we need to understand the relationship between power and time. The key is to recognize that the power (and thus heating effect) of the coils can be related to their time to boil the water individually. The concept involves the following steps:

The power required to boil the water is inversely proportional to the time taken, i.e., if a coil takes less time to boil the water, its power is greater.
Let the power of the first coil be P_1 and it takes 10 minutes to boil the water. Therefore, the energy required to boil the water using the first coil is given by: E = P_1 \times 10.
Similarly, let the power of the second coil be P_2. It takes 40 minutes to boil the same quantity of water, so the energy required is: E = P_2 \times 40.
Since both instances are boiling the same amount of water, the energy E is the same in both scenarios. Therefore, equating both expressions for energy, we have: P_1 \times 10 = P_2 \times 40.
From this, we derive the relation: P_1 = 4P_2.
Now consider both coils are connected in parallel. The total power P_{\text{total}} would be the sum of the individual powers: P_{\text{total}} = P_1 + P_2.
Substitute the value of P_1: P_{\text{total}} = 4P_2 + P_2 = 5P_2.
The time t_{\text{parallel}} taken for the water to boil when both coils are used can be found using the relationship that the energy is constant: E = P_{\text{total}} \times t_{\text{parallel}}. Since the energy E is the same for the quantity of water to boil, equate the energy required using one of the single coils: E = P_1 \times 10 = P_{\text{total}} \times t_{\text{parallel}}.
Substitute P_{\text{total}} = 5P_2 and simplify: 4P_2 \times 10 = 5P_2 \times t_{\text{parallel}}. Hence, t_{\text{parallel}} = \frac{4 \times 10}{5} = 8 minutes.

Thus, when both coils are used together in parallel, the water will boil in 8 minutes. This matches option (a) 8 minutes.

The Correct Option is A

Solution and Explanation

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