The electric potential in a region is given as V = – 4 a r 2 + 3 b , where r is distance from the origin, a and b are constants. If the volume charge density in the region is given by ρ = n a ε 0 , then what is the value of n ?

Step-by-step solution

V = - 4 a r 2 + 3 b E = - d V     d r = 8 a r The small amount of flux d ϕ associated with the closed surface bounded by two spheres of radius r and r + d r is d ϕ = E + d E A + d A - E A d ϕ = E d A + d E A d E = 8 a d r   and d A = 8 π r d r Using Gauss's law ρ × 4 π r 2 d r ε 0 = 8 a d r × 4 π r 2 + 8 a r × 8 π r d r ⇒ ρ   =   24 a ε 0 n   =   24