A spherical cavity of radius 2 m is made with its centre 2 m away from the centre of a uniformly charged solid sphere of radius 5 m and charge density ρ = 8.85 × 10⁻¹² C m⁻³. If the electric field inside the cavity is x/y N C⁻¹, where x and y are the smallest possible integers, find (y − x).
Physics · Electrostatic
JEE Mains Advanced

Question

A spherical cavity of radius 2 m is made with its centre 2 m away from the centre of a uniformly charged solid sphere of radius 5 m and charge density ρ = 8.85 × 10⁻¹² C m⁻³. If the electric field inside the cavity is x/y N C⁻¹, where x and y are the smallest possible integers, find (y − x).

Physics · Electrostatic

G1 Physics · Accepted Answer

Step 1: A spherical cavity in a uniformly charged solid is handled by superposition of a full sphere of density ρ and a negative sphere (the cavity) of density −ρ. Inside the cavity the fields combine to a uniform value E=ρd/(3ε₀), where d is the separation of centres. Step 2: d=2 m and ρ=ε₀ C m⁻³ numerically because ρ=8.85×10⁻¹² and ε₀=8.85×10⁻¹². Thus E=2/3 N C⁻¹. Step 3: x/y=2/3, so y−x=1.