Chapter 18, Problem 15Q

To determine

To explain:

The changes in the magnitude of current in a toaster having the heating element made of Nichrome wire just after it is turned on

Answer to Problem 15Q

Solution:

The magnitude of current in the toaster is decreasing.

Known:

It is known that the temperature coefficient of resistance of Nichrome is 0.00017 per °C. Its resistivity is equal to $1.10\times {10}^{-6}\Omega -m$, which is quite high as compared to that of copper $\left(\rho =1.7\times {10}^{-8}\Omega -m\right)$.

Formula used:

  •The power loss in a resistance (R)is given as

       $P=I^{2}R=\frac{V^2}{R}$

  •The rms current in an ac circuit having resistance R, is given as,

       $I_{rms}=\frac{V_{rms}}{R}$

  •The resistance $\left(R_t\right)$ at temperature $t°C$ is correlated to the reference temperature $t_0°C$ and the resistance at that temperature $\left(R_{t_0}\right)$ as,

         $R_t=R_{t_0}\left[1+\alpha \left(t-t_0\right)\right]$

        where α is the temperature coefficient of resistance of the given material

Explanation of Solution

Nichrome wire is used as a heating element in different devices like hair dryers, ovens, toasters etc. due to its high resistivity. When electric current passes through it, the high resistance of the wire starts providing heat and the process is called, Joule Heating, given by the formula

$P=I^{2}R=\frac{V^2}{R}$. Due to this Joule Heating, the temperature of the Nichrome filament keep on increasing just after the toaster is turned on.

Now, as we know that the temperature coefficient of resistance of Nichrome is 0.00017 per °C, which is ‘positive’, therefore with the increase in the temperature of the element, the resistance of the Nichrome wire keep on increasing, which is given by the formula $R_t=R_{t_0}\left[1+\alpha \left(t-t_0\right)\right]$.

Because the supply voltage $\left(V_{rms}\right)$ to the toaster is assumed to be constant, from the formula $I_{rms}=\frac{V_{rms}}{R}$, we can conclude that the increasing resistance of the Nichrome wire will result in a decreasing current $\left(I_{rms}\right)$, immediately after the toaster is turned on.