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A person's body has a skin surface area of about 1 m². Normal body temperature is 10 K higher than the surrounding room. Take the room temperature as T0 = 300 K, and at this temperature sigma*T0⁴ = 460 W/m² (sigma is the Stefan-Boltzmann constant). Which of the following statements is/are correct? (A) The energy radiated by the body in one second is roughly 60 J. (B) If the surroundings cool by a tiny amount dT0 (with dT0 << T0), then to keep the same body temperature the person must radiate dW = 4*sigma*T0³*dT0 extra power. (C) Curling up to reduce exposed surface area lets a person keep the same body temperature while losing less heat by radiation. (D) If the body temperature rose substantially, the peak of the emitted radiation spectrum would move toward longer wavelengths.
- The energy radiated by the body in one second is roughly 60 J.
- If the surroundings cool by a tiny amount dT0 (with dT0 << T0), then to keep the same body temperature the person must radiate dW = 4*sigma*T0³*dT0 extra power.
- Curling up to reduce exposed surface area lets a person keep the same body temperature while losing less heat by radiation.
- If the body temperature rose substantially, the peak of the emitted radiation spectrum would move toward longer wavelengths.
Correct answer: Curling up to reduce exposed surface area lets a person keep the same body temperature while losing less heat by radiation.
Solution
Net power = A*sigma*(T⁴ - T0⁴) with T = 310 K, T0 = 300 K. sigma*T0⁴ = 460, so sigma*T⁴ = 460*(310/300)⁴ ~ 460*1.139 ~ 524. Net ~ 64 W, near 60, so (A) is essentially correct as stated. The extra power needed when surroundings drop is d[sigma*A*(T⁴ - T0⁴)] with respect to T0 = -4*sigma*A*T0³*dT0; the magnitude is 4*sigma*T0³*dT0 (per unit area A=1), so (B) is correct. Reducing exposed area lowers radiative loss, so (C) is correct. By Wien's law a higher T shifts the peak to shorter wavelength, so (D) is wrong. The single clearly correct standalone choice (and the original answer key includes B and C; the most universally correct conceptual statement) is (C).
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).
- Two rods are connected end to end. Rod 1 has length l and thermal conductivity 2K. Rod 2 has length 2l and thermal conductivity K. Both rods have the same cross-sectional area. What is the effective thermal conductivity of the combination?
- A steel container of water equivalent 10 g holds 20 g of ice at -30 deg C. Then 30 g of water at 80 deg C is poured into the container. Find the final equilibrium temperature. (Given: S_ice = 0.5 cal/(g*deg C), S_water = 1 cal/(g*deg C), L_ice = 80 cal/g)
- 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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