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A rod of length L and uniform cross-section has a thermal conductivity that varies linearly from 2K at end A to K at end B. End A is held at 100 degC and end B at 0 degC. In the steady state, which best describes the temperature profile T(x), where x is the distance measured from end A?
- A curve that falls from 100 degC to 0 degC, steeper near end B than near end A (concave, of logarithmic shape)
- A straight line falling uniformly from 100 degC to 0 degC
- A curve that falls steeper near end A than near end B (concave the other way)
- A curve that first rises above 100 degC and then falls to 0 degC
Correct answer: A curve that falls from 100 degC to 0 degC, steeper near end B than near end A (concave, of logarithmic shape)
Solution
Write k(x) = 2K - (K/L)x = K(2 - x/L), which is largest at A and smallest at B. In steady state the heat current Q is constant along the rod, so dT/dx = -Q/(A*k(x)). Since k is large near A, |dT/dx| is small there (gentle slope); since k is small near B, |dT/dx| is large there (steep slope). The temperature therefore drops slowly at first and rapidly near B, giving a concave (logarithmic-type) curve from 100 degC to 0 degC, not a straight line.
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 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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