An infinitely long cylindrical object with radius R has a charge distribution that depends upon distance r from it-s axis like this - - ar - br 2 r - R- a and b are non zero constant- - is volume charge density- If electric field of the cylinder is zero- then the value of a-b is -n R-4 where the value of n upto two decimal places is

By Gauss Theorem, Net charge inside cylinder = 0 ∫ρ V=0 ⇒∫ (2 π r dr)h ρ = 0 ⇒ ∫^R_O (ar + br^2)rdr = 0 ⇒aR^3/3 + bR^4/4 = 0 ⇒a/b = 3R/4 Hence, value of n is ...

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Byju's Answer

Standard XII

Physics

Electric Field Due to a Sphere and Thin Shell

An infinitely...

Question

An infinitely long cylindrical object with radius $R$ has a charge distribution that depends upon distance $r$ from it's axis like this : $ρ = a r + b r^{2} (r \leq R,$ $a$ and $b$ are non zero constant, $ρ$ is volume charge density). If electric field of the cylinder is zero , then the value of $\frac{a}{b}$ is $\frac{- n R}{4}$ where the value of $n$ (upto two decimal places) is

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Solution

By Gauss Theorem,
Net charge inside cylinder = 0
$\int ρ V = 0$
$\Rightarrow \int (2 π r d r) h ρ = 0 \Rightarrow$ $\int_{O}^{R} (a r + b r^{2}) r d r = 0 \Rightarrow \frac{a R^{3}}{3} + \frac{b R^{4}}{4} = 0 \Rightarrow \frac{a}{b} = - \frac{3 R}{4}$
Hence, value of $n$ is $3.$

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