Question

A solid sphere of radius R has a charge Q distributed in its volume with a charge density ρ = krᵃ, where k and a are constants and r is the distance from centre. If the electric field at r = (R)/(2) is (1)/(8) times that at r = R, the value of a is

• A. \(3\)
• B. \(5\)
• C. \(2\)
• D. 7

Hint:

For volume charge density problems: • Find enclosed charge using integration. • Use Gauss law symmetry.

Answer 1

The Correct Option is B

Step 1: Charge enclosed within radius r: Qᵣ ∝ int₀ʳ krᵃ r² dr ∝ rᵃ+3
Step 2: Electric field inside the sphere: E(r) ∝ (Qᵣ)/(r²) ∝ rᵃ+1
Step 3: Given condition: (E(R/2))/(E(R)) = ((1)/(2))ᵃ+1 = (1)/(8)
Step 4: Solving, a+1 = 3 ⟹ a = 5