An electric toaster uses nichrome for its heating element. When a negllgibly small current pases 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 `2.68A.` 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 elements is the same as room temperature. When the toaster is connected to the supply. Itys 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 stready state will be reachede when temperature will reise no further, and b oth thte reststance 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)/( 2.68A) = 85.8 Omega`
Using the relation
`R_(2) = R_(1) [1 + alpha (T_(2) -T_(1))]`
with `alpha = 1.70 xx10 ^(-4) ^(@)C ^(-1),` we get
`T_(2) - TG_(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 effect due to the current equals heat loss to the surroundings) is `847^(@)C.`