To determine the resistance of a copper coil at [tex]\( 50^{\circ} \text{C} \)[/tex], given that its resistance at [tex]\( 0^{\circ} \text{C} \)[/tex] is [tex]\( 3.35 \Omega \)[/tex] and the temperature coefficient of resistance for copper, [tex]\(\alpha = 4.3 \times 10^{-3} \, ^\circ\text{C}^{-1}\)[/tex], we can use the following formula:
[tex]\[ R_t = R_0 (1 + \alpha \Delta T) \][/tex]
Where:
- [tex]\( R_t \)[/tex] is the resistance at the final temperature (which we need to find).
- [tex]\( R_0 \)[/tex] is the resistance at the initial temperature ([tex]\(0^\circ \text{C}\)[/tex]).
- [tex]\(\alpha\)[/tex] is the temperature coefficient of resistance.
- [tex]\(\Delta T\)[/tex] is the change in temperature ([tex]\( T_{\text{final}} - T_{\text{initial}} \)[/tex]).
### Step-by-Step Solution:
1. Given Values:
- Initial resistance, [tex]\( R_0 = 3.35 \Omega \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 0^\circ \text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = 50^\circ \text{C} \)[/tex]
- Temperature coefficient of resistance for copper, [tex]\(\alpha = 4.3 \times 10^{-3} \, ^\circ\text{C}^{-1}\)[/tex]
2. Calculate the Change in Temperature:
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 50^\circ \text{C} - 0^\circ \text{C} = 50^\circ \text{C} \][/tex]
3. Substitute the values into the resistance formula:
[tex]\[ R_t = 3.35 \Omega \times (1 + 4.3 \times 10^{-3} \times 50) \][/tex]
4. Perform the multiplication and addition inside the parenthesis first:
[tex]\[ 1 + 4.3 \times 10^{-3} \times 50 = 1 + 0.215 = 1.215 \][/tex]
5. Now multiply the initial resistance by this factor:
[tex]\[ R_t = 3.35 \Omega \times 1.215 \][/tex]
6. Calculate the final result:
[tex]\[ R_t = 4.07025 \Omega \][/tex]
Thus, the resistance of the copper coil at [tex]\( 50^\circ \text{C} \)[/tex] is approximately [tex]\( 4.07025 \Omega \)[/tex].
