An electric toaster uses nichrome for its heating element. When a negligibly small current passes through it, its resistance at room temperature `(27.0^(@)C)` is found to be `75.3 Omega`. When the toaster is connected to a 230 V supply, the current settles, after a few seconds, to a steady value of 2.68 A. what is the steady temperature of the nichrome element? The Temperature coefficient of resistance of nichrome averaged over the temperature range involved is `1.70 xx 10^(-4) .^(@)C^(-1)`.

Explanation

When the current through the element is very small, heating effects can be ignored and the temperature `T_(1)` of the element is the same as room temperature. When the toaster is connected to the supply, its initial current will be slightly higher than its steady value of 2.68 A. But due to heating effect of the current, the temperature will rise. This will cause an increase in resistance and a slight decrease in current. In a few seconds, a steady state will be reached when temperature will rise no further, and both the resistance of the element and the current drawn will achieve steady values. The resistance `R_(2)` at the steady temperature `T_(2)` is
`R_(2) = (230 V)/(2.68 A) = 85.8 Omega`
Using the relation
`R_(2) = R_(1) [1 + alpha (T_(2) - T_(1))]`
With `alpha = 1.70 xx 10^(-4) .^(@)C^(-1)`, we get
`T_(2) - T_(1) = ((85.8 - 75.3))/(75.3( xx 1.70 xx 10^(-4)) )= 820^(@)C`
that is, `T_(2) = (820 + 27.0) .^(@)C = 847^(@)C`
Thus, the steady temperature of the heating element (when heating effect due to the current equals heat loss to the surroundings ) is `847^(@)C`