University Physics IV · topic UPIV-01.02
Cavity radiation, Kirchhoff's law, and the pre-Planck laws
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Cavity radiation and the pre-Planck laws: predict a finite change
Cavity (blackbody) radiation is the electromagnetic field in thermal equilibrium with the walls of a sealed enclosure, and its spectrum depends only on temperature — never on the material, the size, or the shape of the cavity. Use the supplied topic model Y_out = 14387.8/max(w*t,1), where Y_out is x = hc/λkT AT CURSOR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. w is Cursor wavelength λ, measured in µm. Keep t (Cavity temperature T) = 1200 K fixed. For this exercise use only 21.6969 µm ≤ w ≤ 178.803 µm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at w = 56.2602 µm and 78.2551 µm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δw across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
x = hc/λkT AT CURSOR against w
Predictions from the supplied Cavity radiation and the pre-Planck laws model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): x = hc/λkT AT CURSOR
Horizontal axis (x): w: Cursor wavelength λ / µm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| w: Cursor wavelength λ / µm | x = hc/λkT AT CURSOR |
|---|---|
| 21.696875 | 0.5526064621441259 |
| 60.973437499999996 | 0.1966402719763558 |
| 100.25 | 0.11959933499584371 |
| 139.52656249999998 | 0.08593226349522755 |
| 178.803125 | 0.06705606142696517 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of w separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At w = 56.2602 µm, substitute into Y_out = 14387.8/max(w*t,1): Y₀ = 0.213114.
- At w = 78.2551 µm, the same fixed-parameter model gives Y₁ = 0.153215.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0598991; the input change is Δw = 21.9949 µm.
- Average sensitivity is ΔY_out/Δw = -0.00272332 1 per µm. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 0.213114, Y₁ = 0.153215; ΔY_out = -0.0598991; average sensitivity = -0.00272332 1 per µm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Cavity radiation and the pre-Planck laws: infer an input with uncertainty
Cavity (blackbody) radiation is the electromagnetic field in thermal equilibrium with the walls of a sealed enclosure, and its spectrum depends only on temperature — never on the material, the size, or the shape of the cavity. Use the supplied topic model Y_out = 14387.8/max(w*t,1), where Y_out is x = hc/λkT AT CURSOR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. w is Cursor wavelength λ, measured in µm. Keep t (Cavity temperature T) = 1200 K fixed. For this exercise use only 21.6969 µm ≤ w ≤ 178.803 µm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.10201362, with a bounded reading uncertainty ±0.0052436. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0157106 µm, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
x = hc/λkT AT CURSOR against w
Predictions from the supplied Cavity radiation and the pre-Planck laws model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): x = hc/λkT AT CURSOR
Horizontal axis (x): w: Cursor wavelength λ / µm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| w: Cursor wavelength λ / µm | x = hc/λkT AT CURSOR |
|---|---|
| 21.696875 | 0.5526064621441259 |
| 60.973437499999996 | 0.1966402719763558 |
| 100.25 | 0.11959933499584371 |
| 139.52656249999998 | 0.08593226349522755 |
| 178.803125 | 0.06705606142696517 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(w) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(21.6969) = 0.552606 and Y_out(178.803) = 0.0670561. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives w_line = 21.6969 + (0.102014 − 0.552606)(178.803 − 21.6969)/(0.0670561 − 0.552606) = 167.492 µm. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives w = 117.532 µm; substituting back gives Y_out = 0.10201362.
- The interpolation error in the input is |w_line − w| = 49.9605 µm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.09677002 and 0.10725722. Sort the roots even if the response decreases: 111.786 µm ≤ w ≤ 123.9 µm.
- The lower and upper input deviations are 5.7459 µm and 6.3686 µm. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer w ≈ 117.532 µm; compatible interval [111.786, 123.9] µm. Endpoint interpolation gives 167.492 µm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.