For a charged spherical ball the electrostatic potential inside the ball varies with r as V=2ar²+b, where a and b are constants and r is the distance from the centre. The volume charge density inside the ball is -λ aε. The value of λ is . Here ε is the permittivity of the medium.

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12. The radial field follows from E=-(dV)/(dr)=-d/(dr)(2ar²+b)=-4ar. Apply Gauss's law to a sphere of radius r: E·4π r²=(q_enc)/ε, so q_enc=ε(-4ar)(4π r²)=-16π aε r³. The enclosed volume is 4/3π r³, so ρ=(q_enc)/(4/3π r³)=(-16π aε r³)/(4/3π r³)=-12aε. The density is uniform, and comparing with ρ=-λ aε gives λ=12.

For a charged spherical ball the electrostatic potential inside the ball varies with r as V=2ar²+b, where a and b are constants and r is the distance from the centre. The volume charge density inside the ball is -λ aε. The value of λ is ______. Here ε is the permittivity of the medium.

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