An infinitely long solid cylinder of radius R with uniform volume charge density `rho` has a spherical cavity of radius `R/2` with its centre on the axis of the cylinder (as shown in the figure)

The magnitude of the electric field at point P whcih is at a distance 2R from the axis o the cylinder is given by `(23rhoR)/(6Kepsilon_(0))`. the value of K is

Answer

16

Explanation

Electric field at P due to cylinder
`E_(1)=(lamda)/(2piepsilon_(0)(2R))=(rhopiR^(2))/(4piepsilon_(0)R)=(rhoR)/(4epsilon_(0))`
………`( :. lamda =Q/h=(rhopi^(2)h)/h=rho pir^(2))`
Electric field at P due to spherical cavity,
`E_(2)=1/(4pi epsilon_(0))(rho . 4/3 pi (R^(3))/8)/((2R)^(2))=(rho R)/((24xx4)epsilon_(0))`
`E_("net")=E_(1)-E_(2)`
`=(rhoR)/(4epsilon_(0))-(rho R)/(96epsilon_(0))`
`(rho R)/(4 epsilon_(0))(1-1/24)`
`=23/96(rho R)/(epsilon_(0))=23/(6xx16)(rho R)/(epsilon_(0))`
`:.K=16`