Question

An electric fan and a heater are marked as $100\,{\text{W}}$, ${\text{220}}\,{\text{V}}$ and $1000\,{\text{W}}$, ${\text{220}}\,{\text{V}}$respectively. The resistance of the heater is:
A. equal to that of fan
B. lesser than that of fan
C. greater than that of fan
D. zero

Answer 1

Use the expression for Joule’s law of heating. Convert the expression for Joule’s law of heating in terms of power, voltage and resistance using Ohm’s law and the equation for the power.

Formula used:
The expression for Joule’s law of heating is
$Q = {i^2}Rt$ …… (1)
Here is the heat developed in the conductor, $i$is the current in the conductor, $R$ is the resistance of the conductor and $t$ is the time.
The expression for Ohm’s law is
$V = iR$ ...... (2)
Here, $V$ is the voltage, $i$ is the current and $R$ is the resistance.
The expression for the power is
$P = \dfrac{Q}{t}$ …… (3)
Here, $P$ is the power, $Q$ is the heat and $t$ is the time.

Complete step by step answer:
Rearrange equation (2) for the current $i$.
$i = \dfrac{V}{R}$
Rearrange equation (3) for heat $Q$.
$Q = Pt$
Substitute $\dfrac{V}{R}$ for $i$ in equation (1).
$Q = {\left( {\dfrac{V}{R}} \right)^2}Rt$
$\Rightarrow Q = \dfrac{{{V^2}}}{R}t$
Substitute $Pt$ for $Q$ in the above equation.
$\Rightarrow Pt = \dfrac{{{V^2}}}{R}t$
$\Rightarrow P = \dfrac{{{V^2}}}{R}$
Rearrange the above equation for resistance $R$.
$\Rightarrow R = \dfrac{{{V^2}}}{P}$
From the above equation, it can be concluded that the resistance is directly proportional to the square of voltage and inversely proportional to the power.
The voltage of the electric fan and the heater are the same which is ${\text{220}}\,{\text{V}}$.
Hence, the resistance of the electric fan and the heater only depend on the power of the electric fan and their relation is inverse.
$\Rightarrow R \propto \dfrac{1}{P}$
Hence, the heater with power has less resistance than the electric fan with lower power.
$\Rightarrow {R_{\text{E}}} > {R_{\text{H}}}$
Here, ${R_{\text{E}}}$ is the resistance of the electric fan and ${R_{\text{H}}}$ is the resistance of the heater.
Hence, the correct option is B.

Note: The energy and the heat are two different physical quantities having the same unit and dimensions. Hence, in the equation for power, the energy can be replaced by the heat.