Question

The resistance of a nichrome wire at $0^\circ C$ is 10 ohm. If the temperature coefficient of resistance is $0.004{^\circ C}^{-1}$, find the resistance at boiling point of the water.

Answer 1

Hint: The boiling point of the water is the temperature at which the molecules get enough energy so as to escape in the atmosphere overcoming the intermolecular forces of attraction and cohesion. Also if the temperature of the wire is increased, then the molecules and atoms inside the wire start vibrating with huge amplitudes which offers more resistance to the electrons to the flow of electricity.

Formula used:
$R = R_{\circ}(1 + \alpha \Delta T)$, where $\alpha$ is called the thermal coefficient of resistance.

Complete answer:
Here $R_\circ$ is the resistance of wire with respect to which temperature change is being taken. Thus $R_\circ = 10\ ohm$ at a temperature of $10^\circ C$. Now, as we know the boiling temperature of water is $100^\circ C$. Hence $\Delta T = 100 - 10 = 90^\circ C$.
Now putting the values in the equation, $R = R_{\circ}(1 + \alpha \Delta T)$, we get;
$R = 10 (1 + 0.004 (90)) = 10 (1 + 0.36) = 10 (1.36) = 13.6\ ohm$
Hence resistance at $100^\circ C\ is \ 13.6\ ohm$.

Additional Information:
The reasons for increase in the resistance of wire are many. This includes the expansion of wire when the wire is heated. Due to the increase in the temperature of the wire, its atoms and molecules start getting far away and hence we observe change in length. Now the resistance of the wire also depends upon length ($R \propto l$), hence the resistance increases with the increase in temperature of the wire.

Note:
One must not get confused that in all problems, we use the absolute temperature for calculation (i.e. temperature in kelvin), but in this question we took in $^\circ C$. This is because the coefficient of thermal expansion has units in $^\circ C ^{-1}$. If it is given in other units, we will take temperature in that unit only.