Question

Two solid spheres of copper with diameter 15cm and 10cm are heated to a temperature of 60 c and 80 degrees celsius kept in contact with each other ? will there be a heat flow between the two spheres, explain?

Answer 1

Given :

The diameter of sphere 1 = $d_1$ = 15 cm

The diameter of sphere 2 = $d_2$ = 10 cm

Temperature = $\Theta _1$ = 60° c

                        $\Theta _2$ = 80°  c

To Find :

The heat flow between two spheres

Solution :

The diameter of sphere 1 = $d_1$ = 15 cm

So, Radius = $r_1$ = $\dfrac{d_1}{2}$ = $\dfrac{15 }{2}$ = 7.5 cm

The diameter of sphere 2 = $d_2$ = 10 cm

So, Radius = $r_2$ = $\dfrac{d_2}{2}$ = $\dfrac{10 }{2}$ = 5 cm

Ratio of heat loss per sec =  H = $\dfrac{d\Theta }{dt}$

i.e  $\dfrac{H_1}{H_2}$ =  $\dfrac{(\dfrac{d\Theta }{dt})_1}{(\dfrac{d\Theta }{dt})_2}$

          = $\dfrac{kA_1 (T-T_0)}{kA_2(T-T_0)}$                         , where A = Area of sphere

          = $\dfrac{A_1}{A_2}$

Or,  $\dfrac{H_1}{H_2}$  =  $\dfrac{A_1}{A_2}$

∵ Area of sphere = 4 π r²        , where  r = radius

So,  $\dfrac{H_1}{H_2}$  =  $\dfrac{A_1}{A_2}$

             = $\dfrac{4\pi r_1^{2} }{4\pi r_2^{2}}$

             = $(\dfrac{r_1}{r_2}) ^{2}$

             = $(\dfrac{7.5}{5}) ^{2}$   =  2.25

So,  $\dfrac{(\dfrac{d\Theta }{dt})_1}{(\dfrac{d\Theta }{dt})_2}$ =  $\dfrac{H_1}{H_2}$ = 2.25

Hence, Heat will flow between two spheres and the ratio of rate of heat change from sphere 1 to sphere 2 is 2.25  Answer