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A rod of length l and cross-section area A has a variable thermal conductivity given by K = α T , where α is a positive constant and T is the temperature in Kelvin. Two ends of the rod are maintained at temperatures T 1 and T 2 ( T 1 > T 2 ) . Heat current flowing through the rod will be -

Options

  1. AA α T 1 2 - T 2 2 3 l
  2. BA α T 1 2 + T 2 2 l
  3. CA α T 1 2 + T 2 2 3 l
  4. DA α T 1 2 - T 2 2 2 l

Correct answer

D. A α T 1 2 - T 2 2 2 l

Step-by-step solution

Heat current, i = - K A d T d X i   d x = - K A   d T i   ∫ 0 l d X = - A α ∫ T 1 T 2 T   d T ⇒ i l = - A α T 2 2 - T 1 2 2 ⇒ i = A α T 1 2 - T 2 2 2 l

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