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Let there be a spherically symmetric cha...

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

`(rho_(0)r)/(4epsilon_(0))((5)/(4)-(r)/(R))`

B

`(4pirho_(0)r)/(3epsilon_(0))((5)/(3)-(r)/(R))`

C

`(rho_(0)r)/(4epsilon_(0))((5)/(3)-(r)/(R))`

D

`(4rho_(0)r)/(3epsilon_(0))((5)/(4)-(r)/(R))`

Text Solution

Verified by Experts

The correct Answer is:
C
`rho(r)={{:(rho_(0)[(5)/(4)-(r)/(R)],0ltrltR),(0" , "rgtR):}`
`E=(int_(0)^(r)rho(r)dV)/(4piepsilon_(0)r^(2))=(Q)/(4piepsilon_(0)r^(2))`
`Q=int_(0)^(r)rho_(0)[(5)/(4)-(r)/(R)]4pir^(2)dr`
`=rho_(0)[int_(0)^(r)5pir^(2)dr-int_(0)^(r)(4pir^(3))/(R)dr]`
`=5rho_(0)pi[(r^(3))/(3)]_(0)^(r)-(4pirho_(0))/(R)[(r^(4))/(4)]=4pirho_(0)[(5r^(3))/(12)-(r^(4))/(4R)]`
By equation (i) -
`implies E=(KQ)/(r^(2))=(1)/(4piepsilon_(0)r^(2))4pirho_(0)[(5)/(12)r^(3)-(r^(4))/(4R)]`
`=(rho_(0)r)/(4epsilon_(0))[(5)/(3)-(r)/(R)]`
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