Question

The volume charge density inside of sphere of radius `R` is given by `rho = rho_(0) (1 - (r )/(R ))` ,where `rho_(0)` is constant and `r` distance from centre.
(a) Find total charge `Q` inside sphere.
Electric field outside sphere at distance `r` from centre.
(b) Electric field outside sphere at distance `r` from centre.
(c) Electric field inside sphere at distance `r` from centre.
(d) At what distance from centre electric field is maximum and its value?
(e ) Sketch `E` versus `r` graph.

Answer 1

To take element on spherical volume , we choose a spherical shell of radius `r` and thickness `dr`
Volume of element
`dV = (4)/(3) pi ( r + dr)^(3) - (4)/(3) pi r^(3) = 4 pi r^(2) dr`
(a) Small charge in small volume
`dq = rho dV = rho_(0) (1 - (r )/(R )) 4 pi r^(2) dr`
`Q = 4 pi rho_(0) int_(0)^(2) (r^(2) = (r^(3))/(R )) dr`
`= 4 pi rho_(0) |(r^(3))/(3) - (r^(4))/(4 R)|_(0)^(R ) = 4 pi rho_(0) ((R^(3))/(3) - (R^(4))/(4R))`
` = (4 pi rho_(0) R^(3))/(12) = (pi rho_(0) R^(3))/(3)`
(b) For a point `(r gt R)`, whole charge is assumed to be concentrated at centre
`E = (1)/(4 pi in_(0)) (Q)/(r^(2)) = (1)/(4 pi in_(0)) .(pi rho_(0) R^(3))/(3r^(2)) = (rho_(0) R^(3))/(12 in_(0)r^(2))`.
`E` versus `r` will be rectangular hyperbola type .
At `r = R , E = (rho_(0) R)/(12 in_(0))`
(c ) For a point `(r lt R)` , change inside sphere of radius `r`
`q = int_(0)^(r ) rho dV = int_(0)^(r ) rho_(0) (1 - (r )/(R)) 4 pi r^(2) dr = 4 pi rho_(0) int_(0)^(2) (r^(2) - (r^(3))/(R )) dr`
`= 4 pi rho_(0) |(r^(3))/(3) - (r^(4))/(4 R)|_(0)^(R ) = 4 pi rho_(0) ((r^(3))/(3) - (r^(4))/(4 R))`
`E = (1)/(4 pi in_(0)) .(q)/(r^(2)) = (1)/(4 pi in_(0)) . 4 pi rho_(0) ((r)/(3) - (r^(2))/(4 R))`
`= (rho_(0))/(in_(0)) ((r)/(3) - (r^(2))/(4 R))`,
`E` versus `r` will be parabola open downward.
At `r = R , E = (rho_(0))/(in_(0)) ((R )/(3) - (R^(2))/(4R)) = ( rho_(0) R)/(12 in_(0))`
(d) If `r gt R` i.e. outside sphere , electric field is decreasing continuously as
`E = (rho_(0) R^(3))/(12 in_(0) r^(2))`
The electric field will be maximum inside sphere `(r lt R)`,
`E = (rho_(0))/(in_(0)) ((r )/(3) - (r^(2))/(4 R))`
For `E` to be maximum
`(dE)/(dr) = (rho_(0))/(in_(0)) ((1)/(3) - (2 r)/(4 R)) = 0 rArr r = (2 R)/(3)`
`E_(max) = (rho_(0))/(in_(0)) [(2R)/(9) - (R )/(9)] = (rho_(0) R)/(9 in_(0))`
(e ) `E` versus `r` graph