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

Concept: Temperature Dependence of Resistance — resistance changes linearly with temperature: $R=R_0(1+\alpha\Delta T)$.

1. Hot resistance from Ohm's law: $R=\dfrac{V}{I}=\dfrac{230}{2.68}\approx 85.82\ \Omega$.
2. $\dfrac{R}{R_0}-1=\dfrac{85.82-75.3}{75.3}\approx 0.1397$.
3. $\Delta T=\dfrac{0.1397}{1.70\times10^{-4}}\approx 821.8\ ^\circ\text{C}$, so $T=27.0+821.8\approx 848.8\ ^\circ\text{C}$.

A metal's resistance rises with temperature. At room temperature the nichrome element has a known (cold) resistance $R_0$; once it settles into steady glowing operation, it draws a fixed current from the $230\ \text{V}$ supply, which fixes its (hot) resistance $R$. Comparing the two through the linear temperature law gives the operating temperature.

Given: $T_0=27.0\ ^\circ\text{C}$, $R_0=75.3\ \Omega$, $V=230\ \text{V}$, $I=2.68\ \text{A}$, $\alpha=1.70\times10^{-4}\ ^\circ\text{C}^{-1}$ (already an average over the temperature range involved, so the linear law can be used directly).

Step 1 — Hot resistance (Ohm's law):

Step 2 — Temperature dependence of resistance:

Step 3 — Substitute (keeping full precision):

Step 4 — Solve for T:

Method: Using the Temperature Dependence of Resistance Formula

This is a direct application of the linear approximation for resistance variation with temperature, given by:

Where:

• $R_T$ = resistance at final temperature $T$
• $R_0$ = resistance at reference temperature $T_0$
• $\alpha$ = temperature coefficient of resistance (assumed constant over the range)

Step 1: Find the steady-state operating resistance

When the toaster is connected to $230\ \text{V}$ and draws $2.68\ \text{A}$, use Ohm’s law:

Step 2: Identify the known quantities

• $R_0 = 75.3\ \Omega$ (at $T_0 = 27.0\ ^\circ\text{C}$)
• $R_T = 85.82\ \Omega$ (at unknown $T$)
• $\alpha = 1.70 \times 10^{-4}\ ^\circ\text{C}^{-1}$

Step 3: Rearrange the formula to solve for $T$

From:

Divide both sides by $R_0$:

Subtract 1:

Divide by $\alpha$:

Finally:

Step 4: Substitute and calculate

First compute the resistance ratio:

Then:

Now:

Step 5: Final answer

Key exam tip: Always check that the temperature rise is reasonable — nichrome heating elements typically operate in the range $800\ – 1100\ ^\circ\text{C}$, so this answer is physically plausible.

Here are the most common mistakes students make on this problem, along with how to avoid each.

1. Using the Wrong Formula for Temperature Dependence

The Mistake:

Students often use the approximate linear formula $R_T = R_0 (1 + \alpha T)$ but forget that $T$ must be the change in temperature ($\Delta T$), not the final temperature itself. They plug in $27^\circ\text{C}$ directly into the formula incorrectly.

How to Avoid:

Always write the correct relation:

Where:

• $R_T$ = resistance at final temperature $T$
• $R_0$ = resistance at initial temperature $T_0$
• $\alpha$ = temperature coefficient of resistance

Key point: The term $(T - T_0)$ is the temperature difference, not the absolute temperature.

2. Confusing $R_0$ with the Operating Resistance

The Mistake:

Some students take $R_0 = 75.3\ \Omega$ as the resistance at the final steady temperature. They then try to solve for $T$ using $R_0$ as the unknown.

How to Avoid:

Clearly label:

• At room temperature ($27^\circ\text{C}$): $R_0 = 75.3\ \Omega$
• At steady operating temperature ($T$): $R_T$ must be calculated from Ohm’s law using the given voltage and current.

3. Incorrectly Calculating $R_T$ from the Circuit

The Mistake:

Using $R = V/I$ but with the wrong values — for example, using the initial current (which is negligibly small) instead of the steady current.

How to Avoid:

Use the steady-state values only:

Calculate carefully:

4. Sign Errors in the Temperature Difference

The Mistake:

When rearranging the formula, students sometimes write $T - T_0 = \frac{R_T - R_0}{\alpha R_0}$ but then forget that $R_T > R_0$ (since resistance increases with temperature for nichrome). They end up with a negative temperature difference.

How to Avoid:

Check physically: nichrome’s $\alpha$ is positive, so if the toaster gets hot, $R_T > R_0$. Therefore $T - T_0$ must be positive. If your calculation gives a negative value, you have swapped $R_T$ and $R_0$ or made a sign error.

5. Forgetting to Add $T_0$ at the End

The Mistake:

After finding $\Delta T = T - T_0$, students stop and give $\Delta T$ as the final answer instead of the actual steady temperature $T$.

How to Avoid:

Always write the final step explicitly:

Here:

Calculate $\Delta T \approx 822^\circ\text{C}$, then:

Final answer: $\mathbf{849\ ^\circ\text{C}}$ (approximately).

Quick Checklist to Avoid All Mistakes

By following this structured approach, you will avoid the common pitfalls and get the correct answer every time.