Chapter 28, Problem 58PQ

(a)

To determine

The power delivered by the power supply for each measurement in Table 28.4.

Answer to Problem 58PQ

The power delivered by the power supply for each measurement in Table 28.4 is shown in Table 1.

Explanation of Solution

Power delivered by the power supply is the same as the power radiated by the bulb. The expression for power is given by,

  $P=I\Delta V$                                                                                                                     (I)

Here, $P$ is the power, $I$ is the current, and $\Delta V$ is the potential difference.

Conclusion:

The power delivered by the power supply for each measurement in Table 28.4 is shown in Table 1.

Table 1

Bulb A (carbon)			Bulb B (tungsten)
$\Delta V\left(\text{V}\right)$	$I\left(\text{A}\right)$	$P\left(\text{W}\right)$	$\Delta V\left(\text{V}\right)$	$I\left(\text{A}\right)$	$P\left(\text{W}\right)$
$10$	$0.10$	$1.0$	$10$	$0.21$	$2.1$
$21$	$0.27$	$5.7$	$21$	$0.30$	$6.3$
$30$	$0.41$	$12$	$30$	$0.38$	$11$
$41$	$0.59$	$24$	$41$	$0.46$	$19$
$50$	$0.75$	$37$	$50$	$0.51$	$26$
$60$	$0.97$	$58$	$60$	$0.56$	$34$
$70$	$1.17$	$82$	$70$	$0.63$	$44$
$80$	$1.36$	$109$	$80$	$0.67$	$54$
$90$	$1.55$	$140$	$91$	$0.71$	$65$
			$101$	$0.77$	$78$
			$110$	$0.79$	$87$

Therefore, the power delivered by the power supply for each measurement in Table 28.4 is shown in Table 1.

(b)

To determine

The temperature of each filament for each entry in the table Table 28.4.

Answer to Problem 58PQ

The temperature of each filament for each entry in the table Table 28.4 is shown in Table 2.

Explanation of Solution

Given that the emissivity of filaments is $1$.

Write the Stefan-Boltzmann equation for the power.

  $P=\sigma \epsilon A{T}^4$                                                                                                                (II)

Here, $\sigma$ is the Stefan-Boltzmann constant, $\epsilon$ is the emissivity, $A$ is the surface area, and $T$ is the temperature.

Write the expression for the surface area of the filament.

  $A=\pi dl$                                                                                                                   (III)

Here, $d$ is the diameter of the filament, and $l$ is its length.

Use equation (III) in (II) and solve for $T$.

  $\begin{array}{c}P=\sigma \epsilon \pi dl{T}^4\\ T={\left(\frac{P}{\sigma \epsilon \pi dl}\right)}^{1/4}\end{array}$                                                                                                         (IV)

Conclusion:

Substitute $0.045\text{mm}$ for $d$, $580\text{mm}$ for $l$, $5.67\times {10}^{-8}{\text{W/m}}^{\text{2}}\cdot {\text{K}}^4$ for $\sigma$, and $1$ for $\epsilon$ in equation (IV) to find the temperature $T$.

  $\begin{array}{c}T={\left(\frac{P}{\pi \left(5.67\times {10}^{-8}{\text{W/m}}^{\text{2}}\cdot {\text{K}}^4\right)\left(1\right)\left(0.045\text{mm}\right)\left(580\text{mm}\right)}\right)}^{1/4}\\ ={\left(\frac{P}{4.6\times {10}^{-12}{\text{W/K}}^4}\right)}^{1/4}\end{array}$

The power corresponding to each measurement is calculated in part (a), so that the above equation can be utilized to compute the temperatures corresponding to each entry. Ii is tabulated in Table 2.

Table 2

Bulb A (carbon)				Bulb B (tungsten)
$\Delta V\left(\text{V}\right)$	$I\left(\text{A}\right)$	$P\left(\text{W}\right)$	$T\left(\text{K}\right)$	$\Delta V\left(\text{V}\right)$	$I\left(\text{A}\right)$	$P\left(\text{W}\right)$	$T\left(\text{K}\right)$
$10$	$0.10$	$1.0$	$680$	$10$	$0.21$	$2.1$	$820$
$21$	$0.27$	$5.7$	$1050$	$21$	$0.30$	$6.3$	$1080$
$30$	$0.41$	$12$	$1300$	$30$	$0.38$	$11$	$1300$
$41$	$0.59$	$24$	$1500$	$41$	$0.46$	$19$	$1400$
$50$	$0.75$	$37$	$1700$	$50$	$0.51$	$26$	$1500$
$60$	$0.97$	$58$	$1900$	$60$	$0.56$	$34$	$1600$
$70$	$1.17$	$82$	$2100$	$70$	$0.63$	$44$	$1800$
$80$	$1.36$	$109$	$2200$	$80$	$0.67$	$54$	$1850$
$90$	$1.55$	$140$	$2300$	$91$	$0.71$	$65$	$1900$
				$101$	$0.77$	$78$	$2000$
				$110$	$0.79$	$87$	$2100$

Therefore, the temperature of each filament for each entry in the table Table 28.4 is shown in Table 2.

(c)

To determine

The plot of $P$ as a function of $T$ for each bulb on the same graph.

Answer to Problem 58PQ

The plot of $P$ as a function of $T$ for each bulb on the same graph is shown in Figure 1.

Explanation of Solution

The computed values of power $P$ and temperature $T$ is shown in Table 2. The plot of plot of $P$ as a function of $T$ for each bulb on the same graph is shown in Figure 1.

Conclusion:

Therefore, the plot of $P$ as a function of $T$ for each bulb on the same graph is shown in Figure 1.