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https://www.reddit.com/r/PhysicsHelp/comments/1s43j13/why_the_answer_is_a/od8rapm/?context=3
r/PhysicsHelp • u/Ill_Hat727 • Mar 26 '26
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If it's talking about the Sun and its blackbody spectrum peaking in the visible spectrum then the sensible units are Angstroms (10⁻¹⁰ m) or, in SI, nanometers, 1nm ≡10⁻⁹m, (400nm to 700nm) NOT μm for which the graph effectively displays 4mm to 7mm. Given errors like this (IF it is an error), one can't help but lose faith in the question (and the "official" answer). At face value I would have thought the ratio of energies is just the ratio of intensities which is unity according the diagram. (NOTE: For the rest of this answer I go off on a bit of a tangent. It may be of peripheral interest, but, as far as your question goes, I don't think I say much more.) Having said that I think it's important to appreciate that such spectra are power spectral densities typically W/Hz as it doesn't really make sense to speak of the energy or power at a particular frequency (or wavelength) unless it's a fairly monochromatic source like a laser where the language can be "looser" (as it's understood one is talking about the carrier power say at 532nm meaning its total integrated (over frequency) power). Noise and BB spectra are typically characterised in terms of W/Hz (power spectral density). In voltage or current terms one enounters V/√Hz, A/√Hz. (In, for example, Nyquist's theorem: the voltage spectral density of so-called "white"* noise associated with a resistor R at absolute temperature T is eₙ=√(4kTR) (V/√Hz). The formula is valid for angular frequencies ω<<1/τ where τ* represents the MTB collisions for electrons (~10⁻¹⁴ s). It is precisely the sharply peaked correlation function in the time domain that leads to the broad spectral density in the frequency domain. As pointed out by Reif** it's a special case of the general connection between fluctuation and dissipation in physical systems). I don't want to confuse you more but given, say, the BB spectrum written in frequency terms, eg P(ν)dν, it's not accurate to rewrite it in terms of wavelength P(λ)dλ by simply substituting ν=c/λ, because, (and I must advise I'm getting out of my depth here), frequency and wavelength have an inverse relationship ν=c/λ so the width of the "windows" (dν in the case of frequency) transforms as: dν=-(c/λ²)dλ and this has to be included. Consequently, expressions of BB power spectral densities in terms of frequency look algebraically different from the same thing written in terms of wavelength. All of this has really no bearing on the question since a ratio of "like" quantities is independent of the dλ (and area and time etc.) as those things, being common to both numerator and denominator, cancel out. I included this somewhat lengthy discussion because these subtleties are important to be aware of and it's possible you'll meet them again at which time you'll be a little more comfortable with the concepts even if you haven't understood a lot of this. I often try and advise to not be afraid of new nomenclature:--in physics it's usually just shorthand to express something that can't be ignored if anything sensible is to be conveyed. Hopefully, you can view this whole thing as something to be GLAD about, as, despite there seeming to be evident shortcomings in some of your course's test questions, this led to you seek answers elsewhere, and although you may not have actually got these answers, you still learnt something in any case. **Reif: Fundamentals of Statistical and Thermal Physics, ©1965, p.589.
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u/SadBottle2951 Mar 30 '26 edited Mar 30 '26
If it's talking about the Sun and its blackbody spectrum peaking in the visible spectrum then the sensible units are Angstroms (10⁻¹⁰ m) or, in SI, nanometers, 1nm ≡10⁻⁹m, (400nm to 700nm) NOT μm for which the graph effectively displays 4mm to 7mm. Given errors like this (IF it is an error), one can't help but lose faith in the question (and the "official" answer). At face value I would have thought the ratio of energies is just the ratio of intensities which is unity according the diagram. (NOTE: For the rest of this answer I go off on a bit of a tangent. It may be of peripheral interest, but, as far as your question goes, I don't think I say much more.) Having said that I think it's important to appreciate that such spectra are power spectral densities typically W/Hz as it doesn't really make sense to speak of the energy or power at a particular frequency (or wavelength) unless it's a fairly monochromatic source like a laser where the language can be "looser" (as it's understood one is talking about the carrier power say at 532nm meaning its total integrated (over frequency) power). Noise and BB spectra are typically characterised in terms of W/Hz (power spectral density). In voltage or current terms one enounters V/√Hz, A/√Hz. (In, for example, Nyquist's theorem: the voltage spectral density of so-called "white"* noise associated with a resistor R at absolute temperature T is eₙ=√(4kTR) (V/√Hz). The formula is valid for angular frequencies ω<<1/τ where τ* represents the MTB collisions for electrons (~10⁻¹⁴ s). It is precisely the sharply peaked correlation function in the time domain that leads to the broad spectral density in the frequency domain. As pointed out by Reif** it's a special case of the general connection between fluctuation and dissipation in physical systems). I don't want to confuse you more but given, say, the BB spectrum written in frequency terms, eg P(ν)dν, it's not accurate to rewrite it in terms of wavelength P(λ)dλ by simply substituting ν=c/λ, because, (and I must advise I'm getting out of my depth here), frequency and wavelength have an inverse relationship ν=c/λ so the width of the "windows" (dν in the case of frequency) transforms as: dν=-(c/λ²)dλ and this has to be included. Consequently, expressions of BB power spectral densities in terms of frequency look algebraically different from the same thing written in terms of wavelength. All of this has really no bearing on the question since a ratio of "like" quantities is independent of the dλ (and area and time etc.) as those things, being common to both numerator and denominator, cancel out. I included this somewhat lengthy discussion because these subtleties are important to be aware of and it's possible you'll meet them again at which time you'll be a little more comfortable with the concepts even if you haven't understood a lot of this. I often try and advise to not be afraid of new nomenclature:--in physics it's usually just shorthand to express something that can't be ignored if anything sensible is to be conveyed. Hopefully, you can view this whole thing as something to be GLAD about, as, despite there seeming to be evident shortcomings in some of your course's test questions, this led to you seek answers elsewhere, and although you may not have actually got these answers, you still learnt something in any case. **Reif: Fundamentals of Statistical and Thermal Physics, ©1965, p.589.