The electric potential in a region is given as `V = -4 ar^2 + 3b`, where r is distance from the origin, a and b are constants. If the volume charge density in the region is given by `rho = n a epsilon_(0)`, then what is the value of n?

Answer

24

Explanation

`V= - 4ar^(2) +3b`
`E= (-dV)/(dr ) = 8 ar `
The small amount of flux d associated with the closed surface bounded by two spheres of radius rand r + dr is
`d phi = (E +m dE ) (A+d A) - EA `
`d phi = Ed A + dEA `
`dE = 8 adr ` and dA = `8 pi r dr `
Using Gauss.s law
`(rho xx 4pi r^(2)dr )/(epsilon_(0)) = 8 adr xx 4pi r^(2) + 8 ar xx 8 pi r dr `
` implies rho = 24 a epsilon_(0)`
`n = 24 `