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Let there be a spherically symmetric charge distribution with

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Let there be a spherically symmetric charge distribution with charge density varying as $ρ (r) = ρ_{0} (\frac{5}{4} - \frac{r}{R})$ upto $r = R$ and $ρ (r) = 0$ for $r > R$ , where $r$ is the distance from the origin. The electric field at a distance $r, (r < R)$ from the origin is given by (a) $\frac{4 π ρ _{0} r}{3 ε _{0}} (\frac{5}{3} - \frac{r}{R})$ (b) $\frac{ρ _{0} r}{4 ε _{0}} (\frac{5}{3} - \frac{r}{R})$ (c) $\frac{4 ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})$ (d) $\frac{ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})$

Views: 5,938 students

Updated on: Jun 18, 2023

Book:

Arihant Physics JEE Main Chapterwise Solutions (2019-2002) (Arihant)

Exam:

JEE Mains 2010

Found 4 tutors discussing this question

Samuel Discussed

Let there be a spherically symmetric charge distribution with charge density varying as

ρ (r) = ρ_{0} (\frac{5}{4} - \frac{r}{R})

upto

r = R

and

ρ (r) = 0

for

r > R

, where

r

is the distance from the origin. The electric field at a distance

r, (r < R)

from the origin is given by (a)

\frac{4 π ρ _{0} r}{3 ε _{0}} (\frac{5}{3} - \frac{r}{R})

(b)

\frac{ρ _{0} r}{4 ε _{0}} (\frac{5}{3} - \frac{r}{R})

(c)

\frac{4 ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})

(d)

\frac{ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})

5 mins ago

Discuss this question LIVE

5 mins ago

Text solutionVerified

Exp. (b) Apply Shell theorem, the total charge upto distance

r

can be calculated as followed

d q = 4 π r^{2} \cdot d r \cdot ρ

= 4 π r^{2} \cdot d r \cdot ρ_{0} [\frac{5}{4} - \frac{r}{R}]

= 4 π ρ_{0} [\frac{5}{4} r^{2} d r - \frac{r ^{3}}{R} d r]

\int d q = q = 4 π ρ_{0} \int_{0}^{r} (\frac{5}{4} r^{2} d r - \frac{r ^{3}}{R} d r]

q = 4 π ρ_{0} [\frac{5}{4} \frac{r ^{3}}{3} - \frac{1}{R} \frac{r ^{4}}{4}]

As electric field intensity,

E = \frac{k q}{r ^{2}} = \frac{1}{4 π ε _{0}} \frac{1}{r ^{2}} \cdot 4 π ρ_{0} [\frac{5}{4} (\frac{r ^{3}}{3}) - \frac{r ^{4}}{4 R}]

E = \frac{ρ _{0} r}{4 ε _{0}} [\frac{5}{3} - \frac{r}{R}]

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Question Text	Let there be a spherically symmetric charge distribution with charge density varying as $ρ (r) = ρ_{0} (\frac{5}{4} - \frac{r}{R})$ upto $r = R$ and $ρ (r) = 0$ for $r > R$ , where $r$ is the distance from the origin. The electric field at a distance $r, (r < R)$ from the origin is given by (a) $\frac{4 π ρ _{0} r}{3 ε _{0}} (\frac{5}{3} - \frac{r}{R})$ (b) $\frac{ρ _{0} r}{4 ε _{0}} (\frac{5}{3} - \frac{r}{R})$ (c) $\frac{4 ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})$ (d) $\frac{ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})$
Updated On	Jun 18, 2023
Topic	Electric Charges and Fields
Subject	Physics
Class	Class 12
Answer Type	Text solution:1 Video solution: 6
Upvotes	522
Avg. Video Duration	8 min

Text solutionVerified

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