Question

Calculate the charge density when a potential function x2 + y2 + z2 is in air(in 10-9 order)​

Answer 1

Given : Potential function, V(x,y, z) = x² + y² + z² is in air (in 10^9 order)

To Calculate : The charge density ( ρ )

solution : we know according to Poisson's equation, |∆²V| = ρ/ε

|∆²V| = δ²V/δx² + δ²V/δy² + δ²V/δz²

= δ²(x² + y² + z²) /δx² + δ²(x² + y² + z²)/δy² + δ²(x² + y² + z²)/δz²

= 2 + 2 + 2 = 6

so, 6 × 10^9 = ρ/ε .........(1)

we know , 1/4πε = k = 9 × 10^9

so, 1/ε = (9 × 10^9 × 4π) putting in equation (1) we get,

6 × 10^9 = ρ × 9 × 10^9 × 4π

⇒ρ = 6/36π = 1/6π units

Therefore the charge density will be 1/6π units.