University Physics IV · unit UPIV-01
Limits of Classical Physics
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.
Extra challenging questions
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The classical synthesis and its limits: infer an input with uncertainty
The classical synthesis is the closed predictive account formed by Newtonian dynamics, Maxwell's electrodynamics and statistical mechanics; its domain of validity is the region in which the ratios v/c, hν/kT and λ/d are all small. Use the supplied topic model Y_out = log(b), where Y_out is log₁₀ (v/c) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; log(x) means log₁₀x. b is Electron speed v, measured in c. Keep s (Cavity size, log₁₀(d / nm)) = 3; t (Temperature T) = 300 K fixed. For this exercise use only 0.106081 c ≤ b ≤ 0.884919 c 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 = -0.23569522, with a bounded reading uncertainty ±0.02739. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000778837 c, 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.
log₁₀ (v/c) against b
Predictions from the supplied The classical synthesis and its limits 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): log₁₀ (v/c)
Horizontal axis (x): b: Electron speed v / c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Electron speed v / c | log₁₀ (v/c) |
|---|---|
| 0.10608125 | -0.9743613714345515 |
| 0.300790625 | -0.521735703898937 |
| 0.4955 | -0.30495634117870585 |
| 0.690209375 | -0.16101914605421552 |
| 0.88491875 | -0.053096602801090194 |
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(b) = 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(0.106081) = -0.974361 and Y_out(0.884919) = -0.0530966. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives b_line = 0.106081 + (-0.235695 − -0.974361)(0.884919 − 0.106081)/(-0.0530966 − -0.974361) = 0.73055 c. 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 b = 0.581172 c; substituting back gives Y_out = -0.23569522.
- The interpolation error in the input is |b_line − b| = 0.149378 c. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -0.26308522 and -0.20830522. Sort the roots even if the response decreases: 0.545651 c ≤ b ≤ 0.619006 c.
- The lower and upper input deviations are 0.0355214 c and 0.0378338 c. 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 b ≈ 0.581172 c; compatible interval [0.545651, 0.619006] c. Endpoint interpolation gives 0.73055 c.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
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.
Mode counting and the ultraviolet catastrophe: infer an input with uncertainty
The Rayleigh–Jeans law is the classical cavity spectrum: the number of electromagnetic modes per unit volume per unit frequency, 8πν²/c³, multiplied by the equipartition energy kT that classical statistics assigns to every one of them. Use the supplied topic model Y_out = 2.998/max(f,0.01), where Y_out is CUT-OFF WAVELENGTH (µm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. f is Cut-off frequency, measured in ×10¹⁴ Hz. Keep t (Cavity temperature T) = 2600 K fixed. For this exercise use only 0.43375 ×10¹⁴ Hz ≤ f ≤ 2.16625 ×10¹⁴ Hz 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 = 2.0113044 µm, with a bounded reading uncertainty ±0.094102 µm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00017325 ×10¹⁴ Hz, 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.
CUT-OFF WAVELENGTH against f
Predictions from the supplied Mode counting and the ultraviolet catastrophe 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): CUT-OFF WAVELENGTH / µm
Horizontal axis (x): f: Cut-off frequency / ×10¹⁴ Hz
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Cut-off frequency / ×10¹⁴ Hz | CUT-OFF WAVELENGTH / µm |
|---|---|
| 0.43374999999999997 | 6.9118155619596555 |
| 0.866875 | 3.4583994232155737 |
| 1.2999999999999998 | 2.3061538461538467 |
| 1.7331249999999998 | 1.7298232960692395 |
| 2.16625 | 1.3839584535487597 |
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(f) = 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(0.43375) = 6.91182 µm and Y_out(2.16625) = 1.38396 µm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives f_line = 0.43375 + (2.0113 − 6.91182)(2.16625 − 0.43375)/(1.38396 − 6.91182) = 1.96963 ×10¹⁴ Hz. 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 f = 1.49057 ×10¹⁴ Hz; substituting back gives Y_out = 2.0113044 µm.
- The interpolation error in the input is |f_line − f| = 0.479057 ×10¹⁴ Hz. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 1.9172024 µm and 2.1054064 µm. Sort the roots even if the response decreases: 1.42395 ×10¹⁴ Hz ≤ f ≤ 1.56374 ×10¹⁴ Hz.
- The lower and upper input deviations are 0.0666219 ×10¹⁴ Hz and 0.0731619 ×10¹⁴ Hz. 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 f ≈ 1.49057 ×10¹⁴ Hz; compatible interval [1.42395, 1.56374] ×10¹⁴ Hz. Endpoint interpolation gives 1.96963 ×10¹⁴ Hz.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Planck's quantised oscillator and the radiation law: infer an input with uncertainty
The Planck distribution gives the energy per unit volume per unit frequency inside a cavity at temperature T, by multiplying the classical count of standing modes by the mean energy hν/(e^(hν/kT) − 1) of an oscillator whose energies are restricted to whole multiples of hν. Use the supplied topic model Y_out = (4799.2*f/t)/(exp(4799.2*f/t)-1), where Y_out is ⟨E⟩ / kT AT PROBE (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. f is Probe frequency ν, measured in ×10¹⁴ Hz. Keep t (Temperature T) = 5800 K fixed. For this exercise use only 0.795 ×10¹⁴ Hz ≤ f ≤ 5.205 ×10¹⁴ Hz 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 = 0.17082732, with a bounded reading uncertainty ±0.016799. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000441 ×10¹⁴ Hz, 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.
⟨E⟩ / kT AT PROBE against f
Predictions from the supplied Planck's quantised oscillator and the radiation law 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): ⟨E⟩ / kT AT PROBE
Horizontal axis (x): f: Probe frequency ν / ×10¹⁴ Hz
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Probe frequency ν / ×10¹⁴ Hz | ⟨E⟩ / kT AT PROBE |
|---|---|
| 0.7949999999999999 | 0.706892633004084 |
| 1.8974999999999997 | 0.41241487995723425 |
| 2.9999999999999996 | 0.22629934438934565 |
| 4.102499999999999 | 0.11785631220721395 |
| 5.204999999999999 | 0.058830788498718166 |
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(f) = 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(0.795) = 0.706893 and Y_out(5.205) = 0.0588308. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives f_line = 0.795 + (0.170827 − 0.706893)(5.205 − 0.795)/(0.0588308 − 0.706893) = 4.44287 ×10¹⁴ Hz. 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 f = 3.4851 ×10¹⁴ Hz; substituting back gives Y_out = 0.17082732.
- The interpolation error in the input is |f_line − f| = 0.957774 ×10¹⁴ Hz. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.15402832 and 0.18762632. Sort the roots even if the response decreases: 3.32513 ×10¹⁴ Hz ≤ f ≤ 3.65969 ×10¹⁴ Hz.
- The lower and upper input deviations are 0.159972 ×10¹⁴ Hz and 0.174586 ×10¹⁴ Hz. 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 f ≈ 3.4851 ×10¹⁴ Hz; compatible interval [3.32513, 3.65969] ×10¹⁴ Hz. Endpoint interpolation gives 4.44287 ×10¹⁴ Hz.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.