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)

A rod of negligible heat capacity has length L, area of cross-section

1.	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)
Answer» Solution :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)`