Question

Volume charge density of a non conducting solid sphere of radius R=30 cm is given as ρ=ρo(1−rR). Electric field inside sphere is minimum at

- r=15 cm
- r=30 cm
- r=10 cm
- r=20 cm

Solution

The correct option is **D** r=20 cm

By Gauss law, ∮→E.→dr=qenclosedϵo

So, E ×4πr2=qenclosedϵ0 ,

where qenclosed=∫ρdV

qenclosed=∫r0ρo(1−rR)4πr2dr

qenclosed=4πρo∫ro(r2−r3R)dr

=4πρo[r33−r44R]ro

qenclosed=4πρo(r33−r44R)E ×4πr2=4πρor2ϵo(r3−r24R)

∴E=ρoϵo(r3−r24R)

For maxima or minima, dEdr=0

13−2r4R=0

r=2R3=2×303 cm=20 cm

By Gauss law, ∮→E.→dr=qenclosedϵo

So, E ×4πr2=qenclosedϵ0 ,

where qenclosed=∫ρdV

qenclosed=∫r0ρo(1−rR)4πr2dr

qenclosed=4πρo∫ro(r2−r3R)dr

=4πρo[r33−r44R]ro

qenclosed=4πρo(r33−r44R)E ×4πr2=4πρor2ϵo(r3−r24R)

∴E=ρoϵo(r3−r24R)

For maxima or minima, dEdr=0

13−2r4R=0

r=2R3=2×303 cm=20 cm

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