Question

The water in an electric kettle begins to boil in ten minutes after being switched on. The wire used as the heating element is modified so that the water begins to boil in $7\,\min $. How can this be achieved?
(A) by increasing the length of the wire
(B) by decreasing the length of the wire
(C) by connecting another identical wire in series
(D) by connecting another identical wire in parallel

Answer 1

Hint The change in the length of the wire can be determined by using the joules law of heating formula, by using the relation which showed by the joules law of heating formula, the change in the length of the wire for the given condition is determined.

Useful formula
The joules law of heating is given by,
$H = \dfrac{{{V^2}t}}{R}$
Where, $H$ is the heat of the wire, $V$ is the potential difference in the wire, $t$ is the time taken and $R$ is the resistance.

Complete step by step solution
Given that,
The time taken by the electric kettle to boil is, $t = 10\,\min $,
The heating element is modified then the time taken is, $t = 7\,\min $.
Now,
The joules law of heating is given by,
$H = \dfrac{{{V^2}t}}{R}\,..............\left( 1 \right)$
In the above equation (1), the resistance of the wire is written as,
$H = \dfrac{{{V^2}t}}{{\left( {\dfrac{{\rho L}}{A}} \right)}}$
By rearranging the terms in the above equation, then the above equation is written as,
$H = \dfrac{{{V^2}tA}}{{\rho L}}$
In the question the relation is given as heat, time taken and the length. So that assume the other terms as constant, then the above equation is written as,
$H \propto \dfrac{t}{L}$
From the above equation, the heat is inversely proportional to the length of the wire. To increase the heat of the wire, then the length of the wire is decreased.

Hence, the option (B) is the correct answer.

Note The heat developed in the wire is directly proportional to the potential difference in the wire, time taken and the heat developed in the wire is inversely proportional to the resistance of the wire. As the potential difference in the wire, time taken increases, then the heat developed also increases.