Question

A heater coil is cut into two equal parts and only one part is used in the heater. The heat generated now will be
A. Doubled
B. Four times
C. One fourth
D. Halved

Answer 1

Find the dependency of heat generated on resistance of the heated coil. Express the resistance in terms of the length. Now, find the dependency of heat generated on the length of the coil. Compare the initial and final case to find the result.

Formula Used :
The resistance of the coil is given by,
$R=\rho \dfrac{l}{A}$
Where,
$\rho$ is the resistivity of the material
$l$ is the length of the coil
$A$ is the area of the coil

The heat generated in the coil is given by,
$Q=\dfrac{{{V}^{2}}t}{R}$
Where,
$V$ is the voltage across the coil
$t$ is the time for which it was on
$R$ is the resistance of the coil

** Complete step-by-step answer :**
The resistance of the coil is given by,
$R=\rho \dfrac{l}{A}$...................(1)
Where,
$\rho$ is the resistivity of the material
$l$ is the length of the coil
$A$ is the area of the coil

Let the length of the coil be l, and the resistance of the coil be R
So, we can write,
$R=\rho \dfrac{l}{A}$
Or, $R\propto l$
Now, if the length of the coil is halved, then the resistance of the coil is,
$R'=\dfrac{R}{2}$
The heat generated by the coil was Q
The heat generated by the coil is given by,
$Q=\dfrac{{{V}^{2}}t}{R}$
Hence, we can write,
$Q\propto \dfrac{1}{R}$
As the new resistance is half the previous resistance, we can write,
$Q'=2Q$
Hence, the heat generated is doubled.
So, the correct choice is (A).

Note :There are several formulas for finding the heat generated. But we have chosen the above formula, because the voltage across the coil is staying the same in both cases. The other well-known formula is,
$Q={{i}^{2}}Rt$

In this formula, the current is not remaining the same. As a result, we did not use this formula.