Question

An electric toaster has resistance of $60 \, \Omega$ at room temperature $(27^\circ \text{C})$. The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A, the temperature attained by the toaster is around: (if $\alpha = 2 \times 10^{-4} / ^\circ \text{C}$)

• A. 1694$^\circ$C
• B. 1235$^\circ$C
• C. 694$^\circ$C
• D. 1667$^\circ$C

Answer 1

Answer: (A)

1694$^\circ$C

Answer 2

Calculate Resistance at Operating Temperature:

Given $V = 220 \, \text{V}$ and $I = 2.75 \, \text{A}$, use Ohm’s law to find the resistance at the elevated temperature:

$R = \frac{V}{I} = \frac{220}{2.75} = 80 \, \Omega$

Use Temperature Coefficient of Resistance Formula:

The relation between the resistance at room temperature $R_0$ and the resistance at temperature $T$ is given by:

$R = R_0 (1 + \alpha \Delta T)$

Substitute $R = 80 \, \Omega$, $R_0 = 60 \, \Omega$, and $\alpha = 2 \times 10^{-4} \ ^\circ \text{C}^{-1}$, where $\Delta T = T - 27$:

$80 = 60 \left(1 + 2 \times 10^{-4} \times (T - 27)\right)$

Solve for $T$:

- Divide both sides by 60:

$\frac{80}{60} = 1 + 2 \times 10^{-4} \times (T - 27)$

- Simplify and isolate $T$:

$\frac{4}{3} - 1 = 2 \times 10^{-4} \times (T - 27)$

$\frac{1}{3} = 2 \times 10^{-4} \times (T - 27)$

$T - 27 = \frac{1}{3 \times 2 \times 10^{-4}} = 1667$

$T = 1667 + 27 = 1694^\circ \text{C}$