A spherically symmetric charge distribution has a volume charge density varying as (r) = ₀ (1 - r R ) for r R , and zero for r > R , where r is the distance from the centre of the sphere and R is its radius. The ratio of the maximum electric field inside the sphere to the electric field at the surface of the sphere is :

Step-by-step solution

According to Gauss's law, the electric flux through a spherical surface of radius r is equal to the enclosed charge divided by ₀ . E d s = Q_ in ₀ E(4 r^2) = 1 ₀ ₀^r (r') 4 r'^2 dr' E(4 r^2) = 4 ₀ ₀ ₀^r (1 - r' R ) r'^2 dr' E(4 r^2) = 4 ₀ ₀ ( r^3 3 - r^4 4R ) E(r) = ₀ ₀ ( r 3 - r^2 4R ) To find the position of maximum electric field, we set dE dr = 0 : dE dr = ₀ ₀ ( 1 3 - 2r 4R ) = 0 r = 2R 3 The maximum electric field is : E_ max = E ( 2R 3 ) = ₀ ₀ ( 2R/3 3 - (2R/3)^2 4R ) = ₀ R ₀ ( 2 9 - 1 9 ) = ₀ R 9 ₀ The elect