Exams › JEE Advanced › Physics
A solid sphere and a cube of the same material and the same volume are heated to the same temperature and allowed to cool under identical surroundings. What is the ratio of the total thermal radiation emitted by the sphere to that emitted by the cube?
- 1: 1
- 4*pi/3: 1
- (pi/6)^(1/3): 1
- (1/2)*(4*pi/3)^(2/3): 1
Correct answer: (pi/6)^(1/3): 1
Solution
Same volume V. Sphere: V = (4/3)*pi*r³ -> r = (3V/(4*pi))^(1/3). Surface area of sphere: Aₛ = 4*pi*r² = 4*pi*(3V/(4*pi))^(2/3). Cube: V = a³ -> a = V^(1/3). Surface area of cube: A_c = 6*a² = 6*V^(2/3). Ratio Aₛ/A_c = 4*pi*(3V/(4*pi))^(2/3) / (6*V^(2/3)) = (4*pi/6) * (3/(4*pi))^(2/3) = (2*pi/3)*(3/(4*pi))^(2/3). Simplifying: (2*pi/3) * 3^(2/3) / (4*pi)^(2/3) = (2*pi/3) * 3^(2/3) / (4^(2/3)*pi^(2/3)) = 2*pi^(1/3)*3^(2/3-1) / (4^(2/3)) =... This simplifies to (pi/6)^(1/3). Since radiation rate is proportional to surface area, ratio is (pi/6)^(1/3): 1.
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.
- A composite wall is built from two layers A and B of equal thickness but made of different materials. The thermal conductivity of layer A is three times that of layer B. In steady-state conditions, the total temperature difference across the entire wall is 36 deg C. What is the temperature difference across layer A alone?
- 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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