NCERT Solutions

We need to determine the final temperature of a heating element based on its change in electrical resistance.

Concept: Temperature coefficient of resistance

The electrical resistance of most metals increases linearly with temperature. This relationship is described by the temperature coefficient of resistance ($\alpha$), which represents the fractional change in resistance per degree change in temperature. The governing formula is:

$$R = R_0 [1 + \alpha(T - T_0)]$$

Where:

• $R$ is the resistance at temperature $T$.
• $R_0$ is the resistance at reference temperature $T_0$.
• $\alpha$ is the temperature coefficient of the material.

Given:

• Reference temperature, $T_0 = 27.0\,^{\circ}\text{C}$
• Initial resistance, $R_0 = 100\,\Omega$
• Final resistance, $R = 117\,\Omega$
• Temperature coefficient, $\alpha = 1.70 \times 10^{-4}\,^{\circ}\text{C}^{-1}$

Solving:

1. Rearrange the linear resistance equation to solve for the temperature difference $\Delta T = T - T_0$:

$$\begin{aligned} R &= R_0 + R_0 \alpha (T - T_0) \\ R - R_0 &= R_0 \alpha (T - T_0) \\ T - T_0 &= \frac{R - R_0}{R_0 \alpha} \end{aligned}$$

2. Substitute the given numerical values into the expression for the temperature change:

$$\begin{aligned} T - 27.0\,^{\circ}\text{C} &= \frac{117\,\Omega - 100\,\Omega}{100\,\Omega \cdotp (1.70 \times 10^{-4}\,^{\circ}\text{C}^{-1})} \\ &= \frac{17}{100 \cdotp 1.70 \times 10^{-4}}\,^{\circ}\text{C} \\ &= \frac{17}{0.017}\,^{\circ}\text{C} \\ &= 1000\,^{\circ}\text{C} \end{aligned}$$

3. Calculate the final temperature $T$:

$$\begin{aligned} T &= 1000\,^{\circ}\text{C} + 27.0\,^{\circ}\text{C} \\ &= 1027.0\,^{\circ}\text{C} \end{aligned}$$

The final temperature of the heating element is:

$$1027.0\,^{\circ}\text{C}$$