Question

The charge following through a resistance R varies with time t as Q = at - bt², where a and b are positive constants. The total heat produced in R is,

a³R/3b

a³R/2b

a³R/b

a³R/6b

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Solution

a³R/6b

Given,
Charge, Q = at - bt² ... (i)
We know that,

Current, I =

dθ over dt

So, equation (i) can be written as,

space space space space space straight I space equals space straight d over dt left parenthesis at minus bt squared right parenthesis rightwards double arrow space straight I space equals space straight a space minus space 2 bt

... (ii)
For maximum value of t, the current is given by,
a-2bt = 0
Therefore,

straight t space equals space fraction numerator straight a over denominator 2 straight b end fraction

...(iii)
Total heat produced (H) can be given as,

straight H space equals space integral subscript 0 superscript straight t straight I squared straight R space dt space space space space equals space integral subscript 0 superscript straight a divided by 2 straight b end superscript left parenthesis straight a minus 2 bt right parenthesis squared space straight R. space dt

open curly brackets because space straight t space equals space fraction numerator straight a over denominator 2 straight b end fraction close curly brackets

space space space space equals space integral subscript 0 superscript straight a divided by 2 straight b end superscript left parenthesis straight a squared plus 4 straight b squared straight t squared minus 4 abt right parenthesis space Rdt straight H space equals space open square brackets straight a squared straight t plus 4 straight b squared straight t cubed over 3 minus fraction numerator 4 abt squared over denominator 2 end fraction close square brackets subscript 0 superscript begin inline style bevelled fraction numerator straight a over denominator 2 straight b end fraction end style end superscript space straight R

On solving the above equation, we get

rightwards double arrow space space straight H space equals space fraction numerator straight a cubed straight R over denominator 6 straight b end fraction

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