The electrostatic potential in a charged spherical region of radius r varies as V=ar³+b, where a and b are constants. The total charge inside a sphere of unit radius is α×π aε₀. The value of α is (the permittivity of vacuum is ε₀)

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(2) -12. The field follows from the potential: Eᵣ=-(dV)/(dr)=-3ar². Apply Gauss's law over the sphere of unit radius: Φ=Eᵣ(4π r²) evaluated at r=1, so Φ=(-3a)(4π)=-12π a. q_enc=ε₀Φ=-12π aε₀. Comparing with α×π aε₀ gives α=-12.

The electrostatic potential in a charged spherical region of radius r varies as V=ar³+b, where a and b are constants. The total charge inside a sphere of unit radius is α×π aε₀. The value of α is (the permittivity of vacuum is ε₀)

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