MPSC 2024 Prelims Paper-I · Question 53 of 100

Correct — option (2), 9550 nm. Every body above absolute zero radiates, and the wavelength at which it radiates most strongly is fixed by its temperature through Wien's displacement law: the peak wavelength multiplied by the absolute temperature equals a constant of about 2.9 thousandths of a metre-kelvin. Put the room temperature of the stem into that relation — the paper writes it as 300k, meaning 300 kelvin — and the peak comes out at roughly 2.9 times ten to the minus three divided by 300, which is about 9.7 micrometres, or something under ten thousand nanometres. Of the four printed choices only 9550 nm lies anywhere near that figure; the exact value from the standard constant is a little above it, but the other three options are wrong by whole orders of magnitude rather than by a few per cent, so the intended answer is unmistakable. The physical meaning is worth holding on to. A body at everyday temperature emits its thermal radiation in the mid to far infrared, well beyond the red end of what the eye can see, which is why objects around us do not glow in the dark and why a camera that can see this radiation must be built with infrared detectors rather than ordinary optics. It is also the reason the atmospheric window between roughly eight and fourteen micrometres matters so much: that band is where the earth's surface, sitting near 300 kelvin, radiates most of its heat, and it is the band that greenhouse gases absorb. Only when a body is heated to a thousand kelvin or more does its peak move far enough toward the visible for it to appear red hot, and only at several thousand kelvin does it look white.

• (1)500 nm — 500 nm sits in the middle of the visible spectrum, in the green, and it is the peak emission wavelength not of a room but of the surface of the Sun, which is at roughly 5,800 kelvin. Wien's law makes the mismatch stark: to peak at 500 nm a body would have to be some twenty times hotter than the room described in the stem. This choice tests whether a candidate has confused the wavelength at which the eye is most sensitive, and the wavelength at which sunlight peaks, with the radiation emitted by an ordinary object at ordinary temperature.
• (3)1500 nm — 1500 nm is in the near infrared, just beyond the visible, and by Wien's law it corresponds to a body at roughly 1,900 kelvin — comparable to the filament of an incandescent lamp rather than to a room. It is a familiar number for another reason entirely, being one of the standard transmission bands of optical fibre communication, and that familiarity is what makes it a plausible-looking choice. It is nonetheless more than six times shorter than the wavelength at which a 300 kelvin body radiates most strongly.
• (4)3000 nm — 3000 nm, or three micrometres, is in the mid infrared and corresponds by Wien's law to a temperature of around 970 kelvin — a body hot enough to glow a dull red. It is the closest of the three wrong choices and the one a candidate might reach for after correctly deciding that the answer must be infrared, having remembered the direction of the relation without carrying the arithmetic through. The calculation is worth doing rather than estimating: dividing the Wien constant by 300 gives close to ten micrometres, not three.

A body in thermal equilibrium emits a continuous spectrum of electromagnetic radiation whose shape depends only on its temperature, and two laws summarise that spectrum. Wien's displacement law fixes where the peak lies: the product of the peak wavelength and the absolute temperature is a constant, about 2.9 times ten to the minus three metre-kelvin, so the hotter the body the shorter the wavelength at which it radiates most strongly. The Stefan-Boltzmann law fixes how much it radiates in total: the power emitted per unit area rises as the fourth power of the absolute temperature, so a modest rise in temperature produces a large rise in emission. Together they explain the everyday sequence of a heated object — dark but radiating in the infrared at room temperature, dull red near a thousand kelvin, orange and then yellow as it climbs, white at several thousand. They also explain the planetary energy balance: the Sun at about 5,800 kelvin radiates mainly in the visible and near infrared, which passes readily through the atmosphere; the earth's surface at about 300 kelvin re-radiates in the far infrared around ten micrometres, which greenhouse gases absorb, and that asymmetry is the mechanism of the greenhouse effect.

MPSC's physics questions on radiation reward one memorised constant and one division. Wien's constant is the only number needed here, and a candidate who carries it can answer any question of this family, in either direction — given a temperature find the peak wavelength, or given a peak wavelength identify the source. The Commission's distractors are built from wavelengths that are individually famous for other reasons: 500 nm for the solar peak and the eye's sensitivity, 1500 nm for optical fibre transmission, 3000 nm as a round mid-infrared figure. Recognising why each number is familiar is what stops it from feeling like the right answer. It is also worth noticing that the printed stem writes the unit of temperature with a lower-case letter, which is a typographical slip rather than a different unit; kelvin is written with a capital K.

• Wien's displacement law states that the wavelength of peak emission multiplied by the absolute temperature of the body equals a constant of about 2.9 times ten to the minus three metre-kelvin.
• At about 300 kelvin, ordinary room temperature, that peak falls near ten micrometres — in the far infrared, far beyond the visible range, which is why objects at room temperature do not glow to the eye.
• The surface of the Sun, at roughly 5,800 kelvin, peaks near 500 nanometres in the visible green, which is why sunlight is rich in visible light and why the eye evolved to be most sensitive there.
• The Stefan-Boltzmann law states that the total power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature, so radiated power rises very steeply with heating.
• The earth's surface radiates most strongly in the eight to fourteen micrometre band, the region absorbed by greenhouse gases, which makes Wien's law the starting point for understanding the greenhouse effect and for the design of thermal imaging equipment.

• Assuming thermal radiation from an everyday object must be visible, when at room temperature the peak lies deep in the infrared
• Confusing the solar peak near 500 nanometres with the peak of a body at room temperature, which is about twenty times longer
• Recognising that the answer must be infrared but not carrying the division through, and so settling for a mid-infrared value several times too short
• Reversing the direction of Wien's law and supposing that a hotter body radiates at a longer wavelength

Radiation questions in MPSC papers test one relation at a time — Wien's law for the position of the peak, Stefan-Boltzmann for the total power, and occasionally Kirchhoff's law relating emission to absorption. The numerical versions are always single-step, so the constant is the only preparation required; the conceptual versions ask why a hot object changes colour as it heats, why the sky or the sun appears as it does, or why the earth's outgoing radiation is trapped by the atmosphere while incoming sunlight is not. Expect the wrong choices to be wavelengths that are individually memorable, which is what makes them feel plausible.

No directly related past PYQ was found.

• practice — not a real PYQ

  According to Wien's displacement law, if the absolute temperature of a black body is doubled, the wavelength at which it emits most strongly will do which of the following ?

  • (a)Double
  • (b)Halve
  • (c)Remain unchanged
  • (d)Increase four times

  Answer(b) Halve — the law states that the peak wavelength multiplied by the absolute temperature is a constant, so the two are inversely proportional and doubling the temperature halves the peak wavelength. This is why a body glows first dull red and then orange, yellow and finally white as it is heated: the peak is moving steadily toward the shorter, bluer wavelengths.

• practice — not a real PYQ

  The surface of the Sun radiates most strongly at a wavelength of about 500 nanometres. Using Wien's displacement law, its surface temperature is closest to which of the following ?

  • (a)1,500 kelvin
  • (b)3,000 kelvin
  • (c)5,800 kelvin
  • (d)15,000 kelvin

  Answer(c) 5,800 kelvin — dividing the Wien constant of about 2.9 times ten to the minus three metre-kelvin by a peak wavelength of 500 nanometres gives roughly 5,800 kelvin. That temperature places the peak in the visible green, which is why sunlight carries so much visible energy, in contrast with a 300 kelvin body whose peak lies near ten micrometres in the far infrared.