Physics-
General
Easy

Question

Five rods of same dimensions are arranged as shown in the figure. They have thermal conductivities K1, K2, K3, K4 and K5. When points A and B are maintained at different temperatures, no heat flows through the central rod if

  1. K subscript 1 end subscript equals K subscript 4 end subscript text and end text   K subscript 2 end subscript equals K subscript 3 end subscript    
  2. K subscript 1 end subscript K subscript 4 end subscript equals K subscript 2 end subscript K subscript 3 end subscript    
  3. K subscript 1 end subscript K subscript 2 end subscript equals K subscript 3 end subscript K subscript 4 end subscript    
  4. fraction numerator K subscript 1 end subscript over denominator K subscript 4 end subscript end fraction equals fraction numerator K subscript 2 end subscript over denominator K subscript 3 end subscript end fraction    

The correct answer is: K subscript 1 end subscript K subscript 4 end subscript equals K subscript 2 end subscript K subscript 3 end subscript


    For no current flow between C and D
    open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A C end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C B end subscriptÞ fraction numerator K subscript 1 end subscript A left parenthesis theta subscript A end subscript minus theta subscript C end subscript right parenthesis over denominator l end fraction equals fraction numerator K subscript 2 end subscript A left parenthesis theta subscript C end subscript minus theta subscript B end subscript right parenthesis over denominator l end fraction
    Þ fraction numerator theta subscript A end subscript minus theta subscript C end subscript over denominator theta subscript C end subscript minus theta subscript B end subscript end fraction equals fraction numerator K subscript 2 end subscript over denominator K subscript 1 end subscript end fraction...(i)
    Also open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A D end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript D B end subscriptÞfraction numerator K subscript 3 end subscript A left parenthesis theta subscript A end subscript minus theta subscript D end subscript right parenthesis over denominator l end fraction equals fraction numerator K subscript 4 end subscript A left parenthesis theta subscript D end subscript minus theta subscript B end subscript right parenthesis over denominator l end fraction
    Þfraction numerator theta subscript A end subscript minus theta subscript D end subscript over denominator theta subscript D end subscript minus theta subscript B end subscript end fraction equals fraction numerator K subscript 4 end subscript over denominator K subscript 3 end subscript end fraction ...(ii)
    It is given that theta subscript C end subscript equals theta subscript D end subscript comma hence from equation (i) and (ii) we get fraction numerator K subscript 2 end subscript over denominator K subscript 1 end subscript end fraction equals fraction numerator K subscript 4 end subscript over denominator K subscript 3 end subscript end fraction Þ K subscript 1 end subscript K subscript 4 end subscript equals K subscript 2 end subscript K subscript 3 end subscript

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