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The mass density of a planet of radius R varies with the distance r from its centre as ρ(r)=ρ₀(1-(r²)/(R²)). Then the gravitational field is maximum at

Asked in JEE Main 3rd Sept 1st Shift 2020 · Field of a sphere and a shell

• (1) r=√3/4R
• (2) r=R
• (3) r=1/(√3)R
• (4) r=√5/9R

Step-by-step solution

Why the other options are wrong

• (1) No consistent treatment of the enclosed mass leads to √3/4R; it is placed close to the true root so that a guess based on eyeballing the density looks reasonable.
• (2) Comes from the habit that the field of a sphere peaks at its surface, which holds for uniform density but not here, where the density falls to zero at the edge.
• (3) Comes from using the local density ρ(r) times the whole volume 4/3π r³ as the enclosed mass instead of integrating, which gives E∝ r(1-(r²)/(R²)) and peaks at R/(√3).

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