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Two objects X and Y have thermal emissivities 0.01 and 0.81 respectively. Their outer surface areas are equal. Both radiate energy at the same rate. The wavelength lambda_Y corresponding to the peak spectral radiance of Y is shifted from the peak wavelength of X by 1.00 micrometres. Given that the temperature of X is 5802 K, find the temperature of Y and its peak wavelength lambda_Y.
- The temperature of Y is 1934 K
- lambda_Y = 1.5 micrometres
- The temperature of Y is 11604 K
- The temperature of Y is 2901 K
Correct answer: The temperature of Y is 1934 K
Solution
Setting emissive power equal: 0.01 * (5802)⁴ = 0.81 * T_Y⁴, so T_Y⁴ = (0.01/0.81) * (5802)⁴ = (1/81) * (5802)⁴, giving T_Y = 5802/3 = 1934 K. By Wien's law, lambda_X = 2898/5802 ~ 0.5 micrometres and lambda_Y = 2898/1934 ~ 1.5 micrometres, so the shift = 1.5 - 0.5 = 1.0 micrometre, consistent with the given data. Options A and B are both correct.
Related JEE Advanced Physics questions
- Match the temperature of a blackbody listed in Group-I to the corresponding statement in Group-II, and select the correct answer.
[Given: Wien’s constant = 2.9 × 10⁻³ m-K and hc/e = 1.24 × 10⁻⁶ V-m]
Group-I:
(P) 2000 K
(Q) 3000 K
(R) 5000 K
(S) 10000 K
Group-II:
(1) The peak wavelength of emitted radiation can cause photoelectron ejection from a metal with a work function of 4 eV.
(2) The peak wavelength of emitted radiation falls within the visible spectrum.
(3) The peak wavelength of emitted radiation produces the broadest central diffraction maximum in a single-slit setup.
(4) The energy radiated per unit area is one-sixteenth of that emitted by a blackbody at 6000 K.
(5) The peak wavelength of emitted radiation is suitable for imaging human bones.
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- A steel rail track of length 1 km was laid at an ambient temperature of 20 deg C with no gaps for thermal expansion. When the temperature rose to 25 deg C, the track buckled and formed an isosceles triangle shape. Given the coefficient of linear expansion of steel is 14 * 10⁻⁶ per K, find the height of the buckle in metres (to the nearest integer).
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- A continuous-flow calorimeter is used in two separate experiments to find the specific heat of a liquid. In the first trial, supplying 60 W raises the liquid temperature by 10 K. In the second trial, the power is doubled to 120 W, but the same 10 K rise is maintained by tripling the flow rate. Assuming heat loss to surroundings is the same in both trials, what is the power lost to the surroundings?
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