Two bodies of masses m1 and m2 and specific heat capacities s1 and s2 are connected by a rod of length ℓ, cross-sectional area A, thermal conductivity K and negligible heat capacity. The whole system is thermally insulated. At time t = 0, the temperature of the firs body is T1 and the temperature of the second body is T2(T2> T1). Find the temperature difference between the two bodies at time t.
Given-
Masses of body = m1 and m2
Specific heat capacities = s1 and s2
Rod of length= ℓ,
Cross-sectional area = A
Thermal conductivity = K
Rate of transfer of heat from the rod is given by –
=
(1)
Where,
= temperature of first and second body.
A= Area of cross section of the
K = thermal conductivity of the
L= length
Heat transfer from the rod in time ∆t –
∆Q=
(2)
Heat loss by the body at temperature T2 is equal to the heat gain by the body at temperature T1.
Heat loss by the body at temperature T2 in time ∆t is –
∆Q=m2s2 
(T2′-T2) (3)
From (1) and (2)
m2s2 
(T2′-T2) =
⇒ T2’=T2- 
∆t
This is the fall in the temperature of the body at temperature T2.
Similarly, rise in temperature of water at temperature T1 is –
T1’=T1+ 
t
Change in the temperature
(T2′-T1′)
= T2- 
∆t - T1- 
t
⇒ {(T2′-T1′)-(T2-T1)}=- 
∆t - 
t
⇒ 
= -
Where 
is the rate of change of temperature difference
⇒
∆T= 
)
Integrating both the sides -
=
⇒ ln |T2-T1| = -
Taking the anti-log
⇒ (T2-T1)=e-λt
Where
λ=