Question

A given quantity of water boils in an electric kettle in 12 min. The length of the heating ele-
ment in the kettle is l. If the same quantity of water is to boil in 10 min on the same mains the length of an identical heating element is

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Answer 1

Answer:

$\dfrac{6}{5}l$

Explanation:

Given:

• t  = time interval in the first case = 12 min
• T = time interval for the second case
• l = length of the heating element required in the first case

Assume:

• h = heat require the amount of water to boil in the first case

• H = heat require the amount of water to boil in the second case
• I = current from the supply

• L = length of the heating element in the second case
• R = resistance of the wire
• $\rho$ = resistivity of the wire
• A = thickness of the wire

Since the wire remains the same, the thickness, resistivity will be the same. Moreover, the heating element is connected to the same mains in both the cases. So the current supply remains the same.

According to the Joule's law of heating, the heat required will be the same in both the cases.

$\therefore h = H\\\Rightarrow I^2Rt = I^2RT\\\Rightarrow I^2(\rho \dfrac{l}{A})t = I^2(\rho \dfrac{L}{A})T\\\Rightarrow lt=LT\\\Rightarrow L = \dfrac{lt}{T}\\\Rightarrow L = \dfrac{l\times 12\ min}{10\ min}\\\Rightarrow L = \dfrac{6l}{5}$

Hence, the length of the heating element should be $\dfrac{6l}{5}$.