Question

If resistance of a conductor at $20^\circ$C is $10\Omega$, then the resistance at $80^\circ$C is $[\alpha = 4 \times 10^{-3} \ ^\circ\text{c}^{-1}]$

Hint:

Always calculate the temperature difference $\Delta T$ first. Be careful with decimal places when multiplying by $\alpha$ as it's usually a small number (e.g., $10^{-3}$ or $10^{-4}$).

Answer 1

Step 1: Understanding the Concept:
The electrical resistance of most conductors increases with temperature. For moderate temperature ranges, this relationship is approximately linear.

Step 2: Key Formula or Approach:
The formula for temperature dependence of resistance is:
\[ R_T = R_0[1 + \alpha(T - T_0)] \]
Where:
$R_T$ = Resistance at temperature $T$
$R_0$ = Resistance at reference temperature $T_0$
$\alpha$ = Temperature coefficient of resistance

Step 3: Detailed Explanation:
Given values:
Reference temperature, $T_0 = 20^\circ$C
Resistance at reference temp, $R_{20} = 10\ \Omega$
Final temperature, $T = 80^\circ$C
Temperature coefficient, $\alpha = 4 \times 10^{-3} \ ^\circ\text{C}^{-1}$
First, find the change in temperature ($\Delta T$):
\[ \Delta T = T - T_0 = 80^\circ\text{C} - 20^\circ\text{C} = 60^\circ\text{C} \]
Now, use the formula to find $R_{80}$:
\[ R_{80} = R_{20} [1 + \alpha \cdot \Delta T] \]
Substitute the values:
\[ R_{80} = 10 \cdot [1 + (4 \times 10^{-3}) \cdot 60] \]
\[ R_{80} = 10 \cdot [1 + 240 \times 10^{-3}] \]
\[ R_{80} = 10 \cdot [1 + 0.24] \]
\[ R_{80} = 10 \cdot [1.24] \]
\[ R_{80} = 12.4\ \Omega \]

Step 4: Final Answer:
The resistance at $80^\circ$C is $12.4\ \Omega$.