A long solid non-conducting cylinder of radius R is charged such that the volume charge density is proportional to r where r is the distance from axis. The electric field E at a distance r (r

Question

A long solid non-conducting cylinder of radius R is charged such that the volume charge density is proportional to r where r is the distance from axis. The electric field E at a distance r (r < R) will depend on r as (A) E propto r (B) E propto r2 (C) E propto (1/r) (D) E propto (1/r2)

Tardigrade Admin · Accepted Answer

ρ propto r; ρ=Kr dQ=ρ V =Kr(2π rdr) × l Using Gaussâs Law, E.2π rl=( displaystyle ∫0r Kr.2π rl.dr/ε0) ⇒ E=K (r2/3ε0) ∴ E propto r2

Q. A long solid non-conducting cylinder of radius $R$ is charged such that the volume charge density is proportional to $r$ where $r$ is the distance from axis. The electric field $E$ at a distance $r (r < R)$ will depend on r as

E $\propto$ r

E $\propto$ r²

E $\propto$ (1/r)

E $\propto$ (1/r²)

$ρ \propto r; ρ = Kr$
$d Q = ρ V$
$= Kr (2 π r d r) \times l$
Using Gauss’s Law,
$E .2 π r l = \frac{\int _{0}^{r} Kr .2 π r l . d r}{ε _{0}}$
$\Rightarrow E = K \frac{r ^{2}}{3 ε _{0}}$
$∴ E \propto r^{2}$

Q. A long solid non-conducting cylinder of radius R is charged such that the volume charge density is proportional to r where r is the distance from axis. The electric field E at a distance r(r<R) will depend on r as

E ∝ r

E ∝ r2

E ∝ (1/r)

E ∝ (1/r2)