A point charge `q` is located at the centre of the spherical layers of uniform isotropic dielectric with relative permitivity `K`.The inside radius of the layer is equal to `a` and the outside radius is equal to `b`.The electrostatic energy inside the dielectirc layer is

Options

A. `(1)/(4piepsilon_(0))(q^(2))/(2K^(2))((1)/(a)-(1)/(b))`
B. `(1)/(4piepsilon_(0))(q^(2))/(K^(2))((1)/(a)-(1)/(b))`
C. `(1)/(4piepsilon_(0))(q^(2))/(2K^(2))((1)/(b)+(1)/(a))`
D. `(1)/(4piepsilon_(0))(q^(2))/(K^(2))((2)/(b)-(1)/(a))`

Answer

A

Explanation

consider a small elemental shell of thickness `dx`
Volume `dv=4pix^(2)(dx)`
Electric field at `x=(1)/(4piepsilon_(0))(q)/(Kx^(2))`
Energy density =`(1)/(2)epsilon_(0)E^(2)`
`=(1)/(2)epsilon_(0)(1)/(4piepsilon_(0))^(2)(q^(2))/(K^(2)x^(4))`
`rArr` Energy in elemental shell is
`=(1)/(2)epsilon_(0)(1)/(4piepsilon_(0))^(2)(q^(2))/(K^(2)x^(4))4pix^(2)dx`
`U=(1)/(4piepsilon_(0))(q^(2))/(2K^(2))int_(a)^(b)(dx)/(x^(2))`
`U=(1)/(4piepsilon_(0))(q^(2))/(2K^(2))((1)/(a)-(1)/(b))`