The electric field inside a spherically symmetric region of radius R is directed radially outward and its magnitude varies with the distance r from the centre as E(r) = E₀ ( r R - r^2 R^2 ) for r R . The volume charge density (r) in this re

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The electric field inside a spherically symmetric region of radius R is directed radially outward and its magnitude varies with the distance r from the centre as E(r) = E₀ ( r R - r^2 R^2 ) for r R . The volume charge density (r) in this region is :

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Correct answer: A. ₀ E₀ R (3 - 4r R )

Using the differential form of Gauss's law in spherical coordinates for a radially symmetric field : E = ₀ 1 r^2 d dr (r^2 E) = ₀ Given E(r) = E₀ ( r R - r^2 R^2 ) , we first find r^2 E : r^2 E = E₀ ( r^3 R - r^4 R^2 ) Now, differentiate with respect to r : d dr (r^2 E) = E₀ ( 3r^2 R - 4r^3 R^2 ) Substitute this back into the Gauss's law equation : 1 r^2 E₀ ( 3r^2 R - 4r^3 R^2 ) = ₀ (r) = ₀ E₀ ( 3 R - 4r R^2 ) (r) = ₀ E₀ R (3 - 4r R ) Answer: ₀ E₀ R (3 - 4r R )

The electric field inside a spherically symmetric region of radius R is directed radially outward and its magnitude varies with the distance r from the centre as E(r) = E₀ ( r R - r^2 R^2 ) for r R . The volume charge density (r) in this region is :

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