Three metallic spherical shells (not concentric) of radii R, 2R and 4R are given charges Q₁ , Q₂ and Q₃ respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges Q₁:Q₂:Q₃ , is

Step-by-step solution

For isolated (not concentric) metallic spherical shells, the charge given to each shell resides entirely on its outer surface. The surface charge density of a spherical shell of radius r and charge Q is given by: = Q 4 r^2 Given that the surface charge densities on the outer surfaces of the three shells are equal: ₁ = ₂ = ₃ Substituting the values of charges and radii for each shell: Q₁ 4 R^2 = Q₂ 4 (2R)^2 = Q₃ 4 (4R)^2 Q₁ R^2 = Q₂ 4R^2 = Q₃ 16R^2 Multiplying by R^2 , we get the ratio of the charges: Q₁ : Q₂ : Q₃ =