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  4. For a charged spherical ball, electrostatic potential inside the ball varies with r as V =2 ar 2+ b. Here, a and b are constant and r is the distance from the center. The volume charge density inside the ball is -λ a ε. The value of λ is ε= permittivity of medium.

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Q. For a charged spherical ball, electrostatic potential inside the ball varies with $r$ as $V =2 ar ^2+ b$.
Here, $a$ and $b$ are constant and $r$ is the distance from the center. The volume charge density inside the ball is $-\lambda a \varepsilon$. The value of $\lambda$ is ____ $\varepsilon=$ permittivity of medium.

JEE MainJEE Main 2023Electrostatic Potential and Capacitance

Solution:

$E =-\frac{ dV }{ dr }=-4 ar \equiv \frac{\rho r }{3 \varepsilon_0} $ (compare)
Result inside uniformly charged solid sphere.
$\rho=-12 a \varepsilon_0$
$\lambda=12$