Two thin metallic spherical shells of radii 20 cm and 30 cm, respectively are placed with their centres coinciding. A material of thermal conductivity `alpha` is filled in the space between the shells. The inner shell is maintained at 300 K and the outer shell at 310 K. If the rate at which heat flows radially through the material is 40 W, find the value of `alpha` (in units of `Js^(-1)m^(-1)K^(-1)`).

Options

A. `(3)/(pi)`
B. `(4pi)/(3)`
C. `(5)/(3pi)`
D. `(pi)/(2)`

Answer

C

Explanation

Given radius of two thin metallic spherical shells,
`r_(1)=20cm=0.2m`
`r_(2)=30cm=0.3m`
temperature of inner shell, `T_(1)=300K`
temperature of outer shcil, `T_(2)=310K`
and rate of heat flow, `H=40V^(2)`
Radial rate of flow of heat through the shell in the steady state,
`H=(dQ)/(dt)=alphaA-(dT)/(dr)=(4pir^(2))(dT)/(dr)`
`(dr)/(r^(2))=(4pialpha)/(H).dT`
Integrate on both sides, we get
`impliesint_(r_(1))^(r_(2))(dr)/(r^(2))=(4pialpha)/(H)int_(T_(1))^(T_(2))dT`
`(r_(2)-r_(1))/(r_(1)r_(2))=(4pialpha)/(H)(T_(2)-T_(1))`
`:.alpha=(H(r_(2)-r_(1)))/(4pir_(1)r_(2)(T_(2)-T_(1)))`
Putting the given values, we get
`alpha=(40(0.3-0.2))/(4pi(0.3)(0.2)(310-300))=(4)/(2.4pi)=(5)/(3pi)`
Hence, the value of `alpha=(5)/(3pi)Js^(-1)m^(-1)K^(-1)`.