Answer:
the thermistor temperature = [tex]325.68 \ ^0 \ C[/tex]
Explanation:
Given that:
A thermistor is placed in a 100 °C environment and its resistance measured as 20,000 Ω.
i.e Temperature
[tex]T_1 = 100^0C\\T_1 = (100+273)K\\\\T_1 = 373\ K[/tex]
Resistance of the thermistor [tex]R_1 =[/tex] 20,000 ohms
Material constant [tex]\beta[/tex] = 3650
Resistance of the thermistor [tex]R_2[/tex] = 500 ohms
Using the equation :
[tex]R_1 = R_2 \ e^{\beta} (\frac{1}{T_1}- \frac{1}{T_2})[/tex]
[tex]\frac{R_1}{ R_2} = \ e^{\beta} (\frac{1}{T_1}- \frac{1}{T_2})[/tex]
Taking log of both sides
[tex]In \ \frac{R_1}{ R_2} = In \ \ e^{\beta} (\frac{1}{T_1}- \frac{1}{T_2})[/tex]
[tex]In \ \frac{R_1}{ R_2} = {\beta} (\frac{1}{T_1}- \frac{1}{T_2})[/tex]
[tex]\frac{ In \ \frac{R_1}{ R_2}}{ {\beta}} = (\frac{1}{T_1}- \frac{1}{T_2})[/tex]
[tex]\frac{1}{T_2} = \frac{1}{T_1} - \frac{ In \ \frac{R_1}{ R_2}}{ {\beta}}[/tex]
[tex]{T_2} = \frac{\beta T_1}{\beta - In (\frac{R_1}{R_2})T}[/tex]
Replacing our values into the above equation :
[tex]{T_2} = \frac{3650*373}{3650 - In (\frac{20000}{500})373}[/tex]
[tex]{T_2} = \frac{1361450}{3650 - 3.6888*373}[/tex]
[tex]{T_2} = \frac{1361450}{3650 - 1375.92}[/tex]
[tex]{T_2} = \frac{1361450}{2274.08}[/tex]
[tex]{T_2} = 598.68 \ K[/tex]
[tex]{T_2} = 325.68 \ ^0 \ C[/tex]
Thus, the thermistor temperature = [tex]325.68 \ ^0 \ C[/tex]
