Question

The temperature dependence of resistance is also quantified by the relation ${\mathbf{R}}_{\mathbf{2}}\mathbf{=}{\mathbf{R}}_{\mathbf{1}}\mathbf{[}\mathbf{1}\mathbf{+}\mathbf{\alpha }\mathbf{(}{\mathbf{T}}_{\mathbf{2}}\mathbf{-}{\mathbf{T}}_{\mathbf{1}}\mathbf{)}\mathbf{]}$ where${\mathbf{R}}_{\mathbf{1}}$ and${\mathbf{R}}_{\mathbf{2}}$ are the resistances at Temperatures ${\mathbf{T}}_{\mathbf{1}}$ and${\mathbf{T}}_{\mathbf{2}}$ , respectively, and $\mathbf{\alpha }$is known as the temperature coefficient of resistance. If a copper wire has a resistance of$\mathbf{55}\mathbf{\Omega }$ at role="math" localid="1655276953435" $\mathbf{20}\mathbf{°}\mathbf{C}$, find the maximum permissible operating temperature of the wire if its resistance is to increase by at most role="math" localid="1655277180970" $\mathbf{20}\mathbf{°}\mathbf{C}$. Take the temperature coefficient at data-custom-editor="chemistry" $\mathbf{20}\mathbf{°}\mathbf{C}$ to be data-custom-editor="chemistry" $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{00382}$.

Short answer

Therefore, the operating temperature is $72.35°\mathrm{C}$.

Step-by-step solution

Given data.

The copper resistance at temperature${\mathrm{T}}_1=20°\mathrm{C}$is ${\mathrm{R}}_{\text{1}}=55\mathrm{\Omega }$.

The resistance increased by 20 %.

The temperature coefficient is $\mathrm{\alpha }=0.00382$.

Determine the formula of resistance dependent on temperature.

Write the formula of resistance.

${\mathbf{R}}_{\mathbf{2}}\mathbf{=}{\mathbf{R}}_{\mathbf{1}}\mathbf{[}\mathbf{1}\mathbf{+}\mathbf{\alpha }\mathbf{(}{\mathbf{T}}_{\mathbf{2}}\mathbf{-}{\mathbf{T}}_{\mathbf{1}}\mathbf{)}\mathbf{]}$ ……. (1)

Here,data-custom-editor="chemistry" ${\mathrm{R}}_1$and data-custom-editor="chemistry" ${\mathrm{R}}_2$are the resistances at Temperatures data-custom-editor="chemistry" ${\mathrm{T}}_1$ anddata-custom-editor="chemistry" ${\mathrm{T}}_2$, respectively, and $\mathrm{\alpha }$is the temperature coefficient of resistance.

Determine the resistance with 20 % increase.

The copper resistance at temperature ${\mathrm{T}}_{\text{1}}=20°\mathrm{C}$ is${\mathrm{R}}_1=55\mathrm{\Omega }$ .

With data-custom-editor="chemistry" $20\%$ increase, the new resistance will be:

$\begin{array}{c}{\mathrm{R}}_2=1.2{\mathrm{R}}_1\\ =1.2\times 55\mathrm{\Omega }\\ =66\mathrm{\Omega }\end{array}$

Determine the temperature.

Substitute $55\Omega$ for data-custom-editor="chemistry" ${\mathrm{R}}_1$, $66\mathrm{\Omega }$for ${\mathrm{R}}_2$, $0.00382$ for $\alpha$, and $20°C$ for data-custom-editor="chemistry" ${\mathrm{T}}_1$ in equation (1).

$\begin{array}{c}66=55[1+0.00382({\mathrm{T}}_2-20)]\\ {\mathrm{T}}_2=\frac{1}{5\times 0.00382}+20\\ {\mathrm{T}}_2=72.35°\mathrm{C}\end{array}$

Therefore, the operating temperature is $72.35°\mathrm{C}$.