Figure shows a system of two concentric spheres of radii `r_(1)" and "r_(2)` at the temperatures `T_(1)" and "T_(2)`, respectively. The radial rate of flow of heat in a substance filled between the two concentric spheres is proportional to

Options

A. `r_(2)-r_(1)`
B. In `(r_(2)/r_(1))`
C. `(r_(2)-r_(1))/(r_(1)r_(2))`
D. `(r_(1)r_(2))/(r_(2)-r_(1))`

Answer

D

Explanation

`because` Radial rate flow of heat in a substance,
`" "(dQ)/(dt)=k4pir^(2)*(d""theta)/(dr)=c` (say) `(because (dQ)/(dt)=kA(dT)/(dr))`
where, `" "` A = area of surface
and `" "` k = thermal conductivity.
or `c/(4pir^(2))dr=kd""theta`
or `cint_(r_(1))^(r_(2))(dr)/(4pir^(2))=kint_(T_(1))^(T_(2))d""theta`
`rArr c/(4pi)((-1)/r)_(r_(1))^(r_(2))=k(theta)_(T_(1))^(T_(2))`
`c/(4pi)(1/r_(1)-1/r_(2))=k(T_(2)-T_(1))`
`rArr " "c=4pik((T_(2)-T_(1)))/(((r_(2)-r_(1))/(r_(1)r_(2))))`
or `" "c=4pik(T_(2)-T_(1))(r_(1)r_(2))/((r_(2)-r_(1)))`
or `" "c propto (r_(1)r_(2))/(r_(2)-r_(1))`
or `" "(dQ)/(dt) propto (r_(1)r_(2))/(r_(2)-r_(1))`
Hence the radial rate of slow of heat in a substance filled between the two concentric sphere is proportional to `((r_(1)r_(2))/(r_(2)-r_(1)))`.