A non-conducting solid sphere has radius R and uniform charge density. A spherical cavity of radius `R/4` is hollowed out of the sphere. The distance between center of cavity is `R/2`. If the charge of the sphere is Q after the creation of the cavity and the magnitude of electric field at the center of the cavity is `E=K((Q)/(4 pi in_(0)R^(2)))`, determined the approximate value of K.

Options

A. 0.32
B. 0.78
C. 0.51
D. 0.45

Answer

C

Explanation

A cavity of radius `R/4` is made inside a non-conducting solid sphere as shown in the figure.

Let `Q_(0)` be the charge, present on the sphere when cavity was absent.
Hence, `E=E_(Q_(0))-E_("cavity")`
Where, `E=` electric field at centre of cavity.
`E_(Q_(0))=` electric field at a distance `R/2` from centre and
`E_("cavity")=` electric field due to cavity itself at its centre.
Hence, `E=(Q_(0))/(4 pi epsilon_(0)R^(3))(R/2)-0`
`rArr E= (Q_(0))/(4 pi epsilon_(0)R^(2))(1/2)` ...(i)
Now, it is given that Q be the charge after the creation of the cavity.
So, `(Q)/(Q_(0))=(4/3 pi (R^(3)-(R/4)^(3)))/(4/3 pi R^(3))`
`:.` Since, for a uniform charged density object, charge `prop` volume
`rArr (Q)/(Q_(0))=(63)/(64)`
`rArr Q_(0)=(Q64)/(63)` ...(ii)
So, from the Eq. (i) and (ii) , we get
`E=(Q)/(4 pi epsilon_(0)R^(2))((64)/(63)xx1/2)`
`=0.5079 [ (Q)/(4 pi epsilon_(0)R^(2)))`
`rArr E=0.51[(Q)/(4 pi epsilon_(0)R^(2))]`
Hence, `K=0.51`
`:.` So, the correct option is (c).