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 in the sphere of unit radius is α × πaε₀. The value of α i $s _____.$
(permittivity of vacuum is ε₀)

Question

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 in the sphere of unit radius is α × πaε₀. The value of α i $s _____.$

(permittivity of vacuum is ε₀)

G1 Physics · Accepted Answer

The radial electric field is E = -dV/dr. Given, V = ar³+b. Therefore, dV/dr = 3ar². Hence, E = -3ar². At unit radius, r = 1, so E = -3a. The electric flux through the spherical surface is Φ = E(4πr²) = (-3a)(4π) = -12πa. By Gauss's law, q = ε₀Φ = -12πaε₀. The question writes q = α × πaε₀. Therefore, α = -12. Correct option: A.

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 in the sphere of unit radius is α × πaε₀. The value of α i s _____. (permittivity of vacuum is ε₀)