Initial length of copper ring is

`L_0 =sqrt(S_0/pi) =sqrt((9.98*10^-4)/pi) =0.0178234 m=1.78234 cm` .

Final Length of copper ring is

`L =sqrt(S/pi) =sqrt(10^-3/pi) =0.0178412=1.78412 cm`

Linear coefficient of thermal expansion for copper is

`alpha =16.6*10^-6 K^(-1)`

The law of linear thermal expansion writes as

`L =L_0*(1+alpha*Delta(T))`

`Delta(T) = (1/alpha)*[(L/L_0)-1]=1/(16.6*10^-6)*(1.78412/1.78234-1)=60.16 deg` `T=T_0 +60.16 =20 +60.16 =80.16 Celsius`

b) coefficient of thermal expansion for Steel is

`alpha_(Fe) =11*10^-6 K^-1`

Now the relation between the lengths is

`L_0*(1+alpha_(Cu)*Delta(T)) =L*(1+alpha_(Fe)*Delta(T))`

`L-L_0 = Delta(T)*(L_0*alpha_(Cu) -L*alpha_(Fe))`

`Delta(T) = (L-L_0)/(L_0*alpha_(Cu) -L*alpha_(Fe)) =178.69 Celsius`

`T = T_0 +178.69 =198.69 Celsius`

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