Question

Let there be a spherically symmetric charge distribution with charge density varying as `rho(r)=rho(5/4-r/R)` upto `r=R`, and `rho(r)=0` for `rgtR`, where r is the distance from the origin. The electric field at a distance r(rltR) from the origin is given by
A. ` (4 pi rho_0r)/(3varepsilon_0) (5/3 -r/R)`
B. ` (4 pi rho_0r)/(3varepsilon_0) (5/3 -r/R)`
C. ` (4 pi rho_0r)/(3varepsilon_0) (5/4 -r/R)`
D. ` (4 pi rho_0r)/(3varepsilon_0) (5/4 -r/R)`

Answer 1

Correct Answer - B
Consider a sherical shell of radius `x` and thickness dx.
Charge on it dq
`dq = prop xx 4 pi x^2 . Dx`
` dq = prop_0 (5/4 - x/R) xx 4 pi x^2 dx`
`q = 4 pi prop _0 int_0^r ((5 x^2)/4 - x^3/R) dx`
`q = 4 prop_0 ( (5r^3)/( 3 xx 4) - r^2/(4R))`
`E = (kq)/r^2 = 1/( 4 pir^2) xx 4 pi prop_0 ((5r^3)/( 3xx4) - r^4/(4R))`
` E = ( prop_0r)/(4 prop_0) (5 /3 - r/R)`.