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A sphere of radius R has a volume densit...

A sphere of radius R has a volume density of charge `rho=kr`, were r is the distance from the centre of the sphere and k is constant . The magnitude of the electric field which exists at the surface of the sphere is given by `(epsilon_(0)=` permittivity of free space)

A

`(4pikR^(4))/(3epsilon_(0))`

B

`(kR)/(3epsilon_(0))`

C

`(4pikR)/(epsilon_(0))`

D

`(kR^(2))/(4pi_(0))`

Text Solution

Verified by Experts

The correct Answer is:
d
Consider a thin spherical shell of radius x and thickness dx as shown in the figure.

Volume of the shell , `dV=4pix^(2)dx`
Let us draw a Gaussian surface od radius `r(rltR)` as shown in the figure above.
Total charge enclosed by the Gaussian surface is
`Q_("in")=int_(0)^(r)rhodV=int_(0)(r)kx4pix^(2)dx=4pikint_(0)^(r)x^(3)dx`
`=4pik[(x^(4))/(4)]_(0)^(r)=pikr^(4)`
According to Gaussa.s law
`E4pir^(2)=(Q_("in"))/(epsilon_(0))orE4pir^(2)=(pikr^(4))/(epsilon_(0))`
At the surface of the sphere , r=R
`thereforeE=(kR^(2))/(4piepsilon_(0))`
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Knowledge Check

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    A
    `E=(sigmaR^(2))/(in_(0)kr^(2))`
    B
    `E=(sigma)/(in_(0)k)`
    C
    `E=R^(2)`
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    A
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