Consider a metallic sphere kept in vacuum. There is only one heat source kept at a distance from the sphere. You need to prove that there will always exist two diametrically opposite points on the sphere where temp would be same irrespective of the position of the heat source.
consider a point A on a greater circle and A' its diametrically opp. point... now let T(x) gives temp of a point x... then consider f(x) = T(x)- T(x') on [A,A'] along the greater circle... then f(x) changes sign in its range as f(A) = - f(A')... so by mean value theorem there exists a point P st f(P) = 0 i.e. temp of P and its opp. point P' are same...
ReplyDeleteCorrect Rik, very well explained
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