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If potential at the surface of earth is assigned zero value,then potential at centre of earth

Answer» For\ r \leq R,\ V(r)= \frac{2}{3} \pi G \rho(r^2-3R^2)

at\ center,\ V(0)= \frac{2}{3} \pi G \rho(-3R^2)=-2 \pi G \rho R^2\\ \\at\ surface,\ V(R)= \frac{2}{3} \pi G \rho(R^2-3R^2)=- \frac{4}{3} \pi G \rho R^2\\ \\if\ we\ assume\ potential\ at\ surface\ is\ 0,\ or\ V'(R)=0,\ then

V'(0)=V(0)-V(R)\\ \\V'(0)= (-2 \pi G \rho R^2)-(- \frac{4}{3} \pi G \rho R^2)\\ \\V'(0)=(\frac{4}{3}-2) \pi G \rho R^2\\ \\V'(0)=\boxed {- \frac{2}{3} \pi G \rho R^2}


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