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A rod of negligible heat capacity has length `L`, area of cross-section `A` and thermal conductivity `K`. The temperature of one end is maintained at `0^(@)C` and the other end is slowly linearly varied from `0^(@)C` to `theta_(0)^(@)C` in time `t_(0)` Assuming no loss of heat through the sides, find the total heat transimitted through the rod in time `t_(0)`

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Resistance of rod `R = (L)/(KA)`
The temperature of left end varies linearly from `0^(@)C` to `theta_(0).^(@)C` in the time `t_(0)`
Temperature of left end at time `t`
`theta = 0 + (theta_(0) - 0)/(t_(0)) t = (theta_(0))/(t_(0)) t`
Heat current `i = (theta - 0)/(R ) = (theta)/(R )`
`(dQ)/(dt) = (theta_(0))/(Rt_(0)) t`
`int dQ = (theta_(0))/(Rt_(2)) int_(0)^(t_(0)) t dt`
`Q = (theta_(0))/(Rt_(0)) (t_(0)^(2))/(2)`
`= (theta_(0) t_(0))/(2 R) = (theta_(0) t_(0) KA)/(2 L)`
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