A sphere of radius R have volume charge density given as ρ (.r.)=Kr for r≤ R =0 for rR If electric field at distance (R/2) from centre is (KR2/N ϵ 0) Then N will be.

Question

A sphere of radius R have volume charge density given as ρ (.r.)=Kr for r≤ R =0 for r>R If electric field at distance (R/2) from centre is (KR2/N ϵ 0) Then N will be. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-iiu05pstcs60c99s.jpg" /> oint overset arrow E⋅ d overset arrow s=(qin/ϵ 0) E4π ((R/2))2=( displaystyle ∫ 0R / 2 ρ 4 π r2 dr/(ϵ )0) ⇒ Eπ R2=( displaystyle ∫ 0R / 2 Kr 4 π r2 dr/ϵ 0)=(K R2/16 ϵ 0)

Q. A sphere of radius $R$ have volume charge density given as
$ρ (r) = Kr$ for $r \leq R$
$= 0$ for $r > R$
If electric field at distance $\frac{R}{2}$ from centre is $\frac{K R ^{2}}{N ϵ _{0}}$ Then $N$ will be.

$\oint E \to \cdot d s \to = \frac{q _{in}}{ϵ _{0}}$
$E 4 π (\frac{R}{2})^{2} = \frac{\int _{0}^{R /2} ρ 4 π r ^{2} d r}{( ϵ ) _{0}}$
$\Rightarrow E π R^{2} = \frac{\int _{0}^{R /2} Kr 4 π r ^{2} d r}{ϵ _{0}} = \frac{K R ^{2}}{16 ϵ _{0}}$

Q. A sphere of radius R have volume charge density given as ρ(r)=Kr for r≤R =0 for r>R If electric field at distance 2R​ from centre is Nϵ0​KR2​ Then N will be.

∮E→⋅ds→=ϵ0​qin​​ E4π(2R​)2=(ϵ)0​∫0R/2​ρ4πr2dr​ ⇒EπR2=ϵ0​∫0R/2​Kr4πr2dr​=16ϵ0​KR2​

Q. A sphere of radius $R$ have volume charge density given as
$ρ (r) = Kr$ for $r \leq R$
$= 0$ for $r > R$
If electric field at distance $\frac{R}{2}$ from centre is $\frac{K R ^{2}}{N ϵ _{0}}$ Then $N$ will be.

$\oint E \to \cdot d s \to = \frac{q _{in}}{ϵ _{0}}$
$E 4 π (\frac{R}{2})^{2} = \frac{\int _{0}^{R /2} ρ 4 π r ^{2} d r}{( ϵ ) _{0}}$
$\Rightarrow E π R^{2} = \frac{\int _{0}^{R /2} Kr 4 π r ^{2} d r}{ϵ _{0}} = \frac{K R ^{2}}{16 ϵ _{0}}$