University Physics V · topic UPV-02.02
Cavity modes, equipartition, and the Planck law
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 modes and the Planck law: predict a finite change
Planck's law u(ν,T) = (8πhν³/c³)/(e^(hν/kT) − 1) is the energy per volume per frequency of thermal radiation: the cavity's mode count multiplied by the mean energy the Boltzmann-weighted ladder Eₙ = nhν allows each mode. Use the supplied topic model Y_out = 1/max(exp(47.99*f/t)-1, 0.000001), where Y_out is PROBE n̄ = 1/(eˣ − 1) (photons). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ; max(a,b) selects the larger value. f is Probe frequency ν, measured in THz. Keep t (Temperature T) = 6000 K fixed. For this exercise use only 193.438 THz ≤ f ≤ 1256.56 THz 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 f = 427.325 THz and 576.162 THz. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δf across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PROBE n̄ = 1/(eˣ − 1) against f
Predictions from the supplied Cavity modes and the Planck 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): PROBE n̄ = 1/(eˣ − 1) / photons
Horizontal axis (x): f: Probe frequency ν / THz
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Probe frequency ν / THz | PROBE n̄ = 1/(eˣ − 1) / photons |
|---|---|
| 193.4375 | 0.27040245345998226 |
| 459.21875 | 0.026062548271025568 |
| 725 | 0.0030404314539991552 |
| 990.78125 | 0.0003618658852608267 |
| 1256.5625 | 0.00004317009447302449 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of f separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At f = 427.325 THz, substitute into Y_out = 1/max(exp(47.99*f/t)-1, 0.000001): Y₀ = 0.0338927 photons.
- At f = 576.162 THz, the same fixed-parameter model gives Y₁ = 0.0100687 photons.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0238239 photons; the input change is Δf = 148.837 THz.
- Average sensitivity is ΔY_out/Δf = -0.000160067 photons per THz. 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 = 0.0338927 photons, Y₁ = 0.0100687 photons; ΔY_out = -0.0238239 photons; average sensitivity = -0.000160067 photons per THz.
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 modes and the Planck law: infer an input with uncertainty
Planck's law u(ν,T) = (8πhν³/c³)/(e^(hν/kT) − 1) is the energy per volume per frequency of thermal radiation: the cavity's mode count multiplied by the mean energy the Boltzmann-weighted ladder Eₙ = nhν allows each mode. Use the supplied topic model Y_out = 1/max(exp(47.99*f/t)-1, 0.000001), where Y_out is PROBE n̄ = 1/(eˣ − 1) (photons). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ; max(a,b) selects the larger value. f is Probe frequency ν, measured in THz. Keep t (Temperature T) = 6000 K fixed. For this exercise use only 193.438 THz ≤ f ≤ 1256.56 THz 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 = 0.0011910066 photons, with a bounded reading uncertainty ±0.00017218 photons. 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 0.106313 THz, 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.
PROBE n̄ = 1/(eˣ − 1) against f
Predictions from the supplied Cavity modes and the Planck 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): PROBE n̄ = 1/(eˣ − 1) / photons
Horizontal axis (x): f: Probe frequency ν / THz
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Probe frequency ν / THz | PROBE n̄ = 1/(eˣ − 1) / photons |
|---|---|
| 193.4375 | 0.27040245345998226 |
| 459.21875 | 0.026062548271025568 |
| 725 | 0.0030404314539991552 |
| 990.78125 | 0.0003618658852608267 |
| 1256.5625 | 0.00004317009447302449 |
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(193.438) = 0.270402 photons and Y_out(1256.56) = 0.0000431701 photons. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives f_line = 193.438 + (0.00119101 − 0.270402)(1256.56 − 193.438)/(0.0000431701 − 0.270402) = 1252.05 THz. 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 = 841.944 THz; substituting back gives Y_out = 0.0011910066 photons.
- The interpolation error in the input is |f_line − f| = 410.105 THz. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.0010188266 photons and 0.0013631866 photons. Sort the roots even if the response decreases: 825.083 THz ≤ f ≤ 861.445 THz.
- The lower and upper input deviations are 16.8603 THz and 19.501 THz. 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 ≈ 841.944 THz; compatible interval [825.083, 861.445] THz. Endpoint interpolation gives 1252.05 THz.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.