Chapter 17, Problem 45P

(a)

To determine

The, numerical value of $R_0$ and $B$ .

Explanation of Solution

Given:

The temperature dependence of resistance of a thermistor is $R=R_0{e}^{B/T}$ . The value of $R$ is $7360\Omega$ at ice point and $153\Omega$ at steam point.

Formula used:

Write the expression of resistance for thermistor.

  $R=R_0{e}^{B/T}$   ........ (1)

Here, $R$ is the resistance of the thermistor, $R_0$ is a empirical constant, $B$ is a empirical constant and $T$ is the operating temperature of the thermistor.

Calculation:

Substitute $7360\Omega$ for $R$ and $273\text{K}$ for $T$ in the expression (1).

  $7360\Omega =R_0{e}^{B/273\text{K}}$   ........ (2)

Substitute $153\Omega$ for $R$ and $373\text{K}$ for $T$ in the expression (1).

  $153\Omega =R_0{e}^{B/373\text{K}}$   ........ (3)

Divide expression (2) by (3).

  $\begin{array}{c}\frac{7360\Omega }{153\Omega }=e^{\left(B/273\text{K}\right)-\left(B/373\text{K}\right)}\\ 48.1\Omega =e^{\left(B/273\text{K}\right)-\left(B/373\text{K}\right)}\end{array}$

Take logarithm of the above expression.

  $\ln \left(48.1\right)=B\left(\frac{1}{273}-\frac{1}{373}\right){\text{K}}^{\text{-1}}$

Simplify the above expression for $B$ .

  $\begin{array}{c}B=\frac{\ln \left(48.1\right)}{\left(\frac{1}{273}-\frac{1}{373}\right){\text{K}}^{\text{-1}}}\\ =3.95\times {10}^3\text{K}\end{array}$

Rearrange the expression (2).

  $R_0=\frac{7360\Omega }{e^{B/273\text{K}}}$

Substitute $3.95\times {10}^3\text{K}$ for $B$ in the above expression.

  $\begin{array}{c}{R}_0=\frac{7360\Omega }{e^{\left(3.95\times {10}^3\text{K}\right)/273\text{K}}}\\ =3.9\times {10}^{-3}\Omega \end{array}$

Conclusion:

Thus, the value of $B$ is $3.95\times {10}^3\text{K}$ and the value of $R_0$ is $3.9\times {10}^{-3}\Omega$ .

(b)

To determine

The, resistance of the thermistor at $98.6°\text{F}$

Explanation of Solution

Given:

The temperature dependence of resistance of a thermistor is $R=R_0{e}^{B/T}$ . The value of $R$ is $7360\Omega$ at ice point and $153\Omega$ at steam point.

Formula used:

Write the expression of resistance for thermistor.

  $R=R_0{e}^{B/T}$

Here, $R$ is the resistance of the thermistor, $R_0$ is a empirical constant, $B$ is a empirical constant and $T$ is the operating temperature of the thermistor.

Calculation:

Substitute $3.95\times {10}^3\text{K}$ for $B$ , $3.9\times {10}^{-3}\Omega$ for $R_0$ and $310\text{K}$ for $T$ in the above equation.

  $\begin{array}{c}R=\left(3.9\times {10}^{-3}\Omega \right)e^{\left(3.95\times {10}^3\text{K}\right)/\text{310}\text{K}}\\ =1.32\times {10}^3\Omega \\ =\left(1.32\times {10}^3\Omega \left(\frac{\text{1}\text{kΩ}}{{\text{10}}^{\text{3}}\text{Ω}}\right)\right)\\ =1.32\text{kΩ}\end{array}$

Conclusion:

Thus, the value of resistance at $98.6°\text{F}$ is $1.32\text{kΩ}$ .

(c)

To determine

The, rate of change of temperature at ice point and steam point.

Explanation of Solution

Given:

The temperature dependence of resistance of a thermistor is $R=R_0{e}^{B/T}$ . The value of $R$ is $7360\Omega$ at ice point and $153\Omega$ at steam point.

Formula used:

Write the expression of resistance for thermistor.

  $R=R_0{e}^{B/T}$

Differentiate both sides of the above expression with respect to $T$ .

  $\begin{array}{c}\frac{dR}{dT}=\frac{d}{dT}\left(R_0{e}^{B/T}\right)\\ =-\frac{RB}{T^2}\end{array}$   ........ (4)

For ice point.

Substitute $7360\Omega$ for $R$ , $3.95\times {10}^3\text{K}$ for $B$ and $273\text{K}$ for $T$ in the expression (4).

  $\begin{array}{c}{\left(\frac{dR}{dT}\right)}_{\text{ice-point}}=-\frac{\left(7360\Omega \right)\left(3.95\times {10}^3\text{K}\right)}{273\text{K}}\\ =-389\Omega /\text{K}\end{array}$

For steam point.

Substitute $153\Omega$ for $R$ , $3.95\times {10}^3\text{K}$ for $B$ and $373\text{K}$ for $T$ in the expression (4).

  $\begin{array}{c}{\left(\frac{dR}{dT}\right)}_{\text{steam-point}}=-\frac{\left(153\Omega \right)\left(3.95\times {10}^3\text{K}\right)}{373\text{K}}\\ =-4.34\Omega /\text{K}\end{array}$

Conclusion:

Thus, the rate of change of resistance at ice point is $-389\Omega /\text{K}$ and at the steam point is $-4.34\Omega /\text{K}$ .

(d)

To determine

The, temperature sensitivity of the thermistor.

Explanation of Solution

Given:

The temperature dependence of resistance of a thermistor is $R=R_0{e}^{B/T}$ . The value of $R$ is $7360\Omega$ at ice point and $153\Omega$ at steam point.

Introduction:

The resistance of a thermistor is a function of temperature. The resistance of a thermistor changes with respect to the temperature.

A thermistor is more sensitive at lower temperature.

Write the expression of resistance of a thermistor as a function of temperature.

  $R=R_0{e}^{B/T}$

As the temperature decreases the thermistor becomes more sensitive.

Conclusion:

Thus, a thermistor becomes more sensitive at lower temperature.