An infinite cylinder of radius R has a linear charge density λ. The volume charge density (C/m3) within the cylinder (r≤R) is rho(r)=rho0rR, where rho0 is a constant to be determined.

ρ₀ = 3λ/(2πR⁴)

Explanation:

If we take an elemental part of the cross sectional Area of the infinitely long cylinder.

The small elemental part has a small length dr in the cylinder's radial direction such that the cross sectional Area of the elemental part = 2πr dr

The linear density over the entire radial length of the cylinder will be equal to the sum of (volume charge density × elemental cross sectional Area)

That is,

λ = Σ ρ₀rR × 2πr dr

summing from 0 to R

λ = ∫ᴿ₀ (ρ₀rR × 2πr) dr

λ = ∫ᴿ₀ ρ₀R × 2πr² dr

λ = [ρ₀R × 2πr³/3 ]ᴿ₀

λ = (2π ρ₀R/3) [ r³ ]ᴿ₀

λ = (2π ρ₀R/3) [ R³ ]

λ = 2π ρ₀R⁴/3

ρ₀ = 3λ/(2πR⁴)

answer: the exterior temperature of commercial jet planes as they leave the stratosphere and return to the ground increases steadily.

explanation:

the lower most layer of atmosphere is troposphere which extends from 0 to 12 km above the surface of the earth. the temperature initially decreases as distance is increased from the earth's surface. commercial jets fly in lower part of the stratosphere. the stratosphere extends upto 50 km. the temperature in stratosphere starts rising again due to presence of ozone which traps the uv rays.

thus, when the planes leave the stratosphere and return to the ground, the exterior temperature increases steadily.