# Problem 4

By Jaan Kalda (TalTech).

The space between two parallel infinite metallic plates is filled with homogeneous material of heat conductivity $\kappa$. The heat conductivity of the metallic plates is much larger than $\kappa$, the distance between the plates is $L$. One of the plates is kept at temperature $T_1$, and the other – at temperature $T_2$. There is a spherical cavity of radius $R$ in the medium between the plates, at equal distance $L/2$ from the both plates; assume that $R\ll L$ . Initially, the cavity is filled with a gas of negligibly small heat conductivity (much smaller than $\kappa$). By how much will change the total heat flux $P$ between the plates (measured in watts) if the spherical cavity is filled with a material of very large heat conductivity (much larger than $\kappa$)?

Please submit the solution of this problem via e-mail to physcs.cup@gmail.com. First hints will appear here on 23rd February 2020, but after the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab. The final deadline for the Problem 4 is 23:59 GMT, 31st May.