Two steel plates plates are soldered on two sides of a copper plate .What tensions will arise in the plates if the temperature is increaased by `t^(@)C` ? All the plates have the same cross- sections the coefficents of expansion ofcopper and steel are `alpha_(c)and alpha_(s)` and thier Young 's modulus . are `Y_(c) andY_(s) `repectively Aera of rach interface +A.
[HInt : The net expansion (thermai + elastic ) is the same for the all the plates .THe tensile force on each steel plate is half the tensile force onthe cooper plate.]

Explanation

KEY IDEAS
Imagine if we have kept the three plates separately , the two steel plates would have expected equally , but more than the copper plate .But since they are rigidly connected, the steel plates expansion will be hindered by the copper plate that will expand more because of stretched by the steel plates.
Since the coefficient of linear thermal expansion of copper `alpha_c` is greater than of steel `alpha_s` , the increase in temperature will lead to compression of the copper plate and development of tension in steel plate. Thus , to conclude the steel plates will be experiencing compressive stress and the copper plates will be experiencing tensile stress. Because they are rigidly joined , the relative elongation of all the three plates are the same .
Calculations : Denoting the compressive force acting on the copper plate from the sides of the steel plates by F, we shall have for the relative elongation of the copper plate:
`(Deltal)/l=alpha_ct-F/(AY_c)`
Both steel plates are subjected to the tensile force from the side of the copper plates , Upon equating the relative elongation of the plants , we obtain .
`alpha_ct-F/(AY_c)=alpha_st+F/(2AY_s)`
Hence , `F=(2AY_sY_c(alpha_c-alpha_s)t)/(2Y_s+Y_c)`
Note : If `alpha_c = alpha_s` then we get force as zero , that is what we should expect , as both plates will have same expansion and they will not be pulling each other .