P vs T Assignment: Radiated Power as a Function of Temperature

Physics 1CC3 Team

P vs T Assignment: Radiated Power as a Function of Temperature

P vs T Assignment: Radiated Power as a Function of Temperature

For an ideal blackbody in thermal equilibrium, theory predicts that the power (P) radiated by the object varies as the 4th power of its temperature, i.e.

$P \propto T^{4}$

The filament of a light bulb in thermal equilibrium loses most of its energy to radiation and so might be expected to behave in a similar way. The temperature of a tungsten filament can be measured through its resistance, 𝑅, by

T = 130.7 + 189.8 (\frac{R}{R_{o}}) - 1.327 {(\frac{R}{R_{o}})}^{2} [^{\circ} K]

(1)

where 𝑅𝑜 is the room temperature resistance of the filament. By adding an ammeter and a voltmeter to the bulb’s circuit, you can measure both the resistance of the bulb ( $𝑅 = 𝑉 ⁄ 𝐼$ ) and the power expended ( $𝑃 = 𝐼 𝑉$ ).

Part 1: Determination of R_o

a) Summarize the data from Measurement Set #1 (measured voltage, measured current, and calculated resistance) with uncertainties for each, and present it in a table in best form. The specs for the orange and blue meters are available here.

b) Plot a graph (with uncertainties) showing your calculated 𝑅 as a function of voltage. The graph should be suitably sized and scaled, and show data points with uncertainties.

c) Use the graph to provide a realistic estimate of 𝑅𝑜, with uncertainty.

Part 2: Power vs Temperature measurement

a) Summarize the data from Measurement Set #2 (measured voltage and current, calculated resistance, temperature, and power), with uncertainties, and present it in a table in best form.

b) Plot a graph (with uncertainties) showing your calculated 𝑃 as a function of 𝑇. The graph should be suitably sized and scaled, and show data points with uncertainties.

Part 3: Ln-Ln Analysis

a) Create a separate table of Ln(T) and Ln(P), with uncertainties.

b) Graph the Ln-Ln data. Data points should be shown with uncertainties. Your graph should be suitably sized and scaled so that all relevant detail is visible. A trendline and descriptive equation should appear on the graph.

c) Estimate the exponent, with uncertainty, for the P vs T function. Comment on your results.

Propagation of Uncertainties

The most common mistake in propagation occurs during multiplication or division. Take as an example the calculation of the uncertainty on the power ( $𝑃 = 𝐼 𝑉$ ) from a measured current, 𝐼 ± ∆𝐼, and a measured voltage, 𝑉 ± ∆𝑉 . You are most likely to calculate the relative uncertainty correctly as

\frac{Δ P}{P} = \sqrt{{(\frac{Δ V}{V})}^{2} + {(\frac{Δ I}{I})}^{2}}

(2)

But… this is not the absolute uncertainty on 𝑃. (Among other things, this quantity is dimensionless – it has no units.) To complete the calculation, you need to multiply by 𝑃, i.e.

Δ P = P \sqrt{{(\frac{Δ V}{V})}^{2} + {(\frac{Δ I}{I})}^{2}}

(3)

In the temperature calculation, the uncertainty on the temperature should be calculated using the function argument

i.e. if 𝑦 = 𝑓(𝑥) , then

Δ y = \frac{d y}{d x} Δ x

(4)

where ∆𝑥 is the uncertainty on 𝑥. In your case, the temperature is a function of two independent variables, R and Ro, so the uncertainty should be calculated as

Δ T = \sqrt{{(\frac{δ T}{δ R} Δ R)}^{2} + {(\frac{δ T}{δ R_{o}} Δ R_{o})}^{2}}

(5)

Provide a sample of your uncertainty calculations.

P vs T Assignment: Radiated Power as a Function of Temperature

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