The electrostatic potential in a charged spherical region of radius r varies as V=a r³+b , where a and b are constants. The total charge in the sphere of unit radius is a _ o . The value of is _ _ _ _ . (permittivity of vacuum is ₀ )

Step-by-step solution

From V = ar^3 + b , the electric field is E = -dV/dr = -3ar^2 . By Gauss's law: E(r) 4 r^2 = Q(r)/ ₀ , so -3ar^2 4 r^2 = Q(r)/ ₀ . This gives Q(r) = -12 ar^4 ₀ . At r = 1 : Q(1) = -12 a ₀ . Since Q(1) = a ₀ , we have = -12