A solid sphere of radius has a charge distributed in its volume with a charge density where and are constants and is the distance from its centre. If the electric field at is times that at find the value of . [2009]

Question

A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $ρ = k r^{a},$ where $k$ and $a$ are constants and $r$ is the distance from its centre. If the electric field at $r = \frac{R}{2}$ is $\frac{1}{8}$ times that at $r = R,$ find the value of $a$ . [2009]

Smart Achievers · Accepted Answer

(2)

Let there be a spherical shell of radius $x$ and thickness $d x$ . Volume of this shell, $V = 4 π x^{2} (d x)$ . Charge enclosed in this shell

$d q = (4 π x^{2}) d x \times k x^{a}$

$or,$ $d q = 4 π k x^{2 + a} d x$

$For r = R :$

Total charge enclosed in the sphere of radius $R$

$Q = \int_{0}^{R} 4 π k x^{2 + a} d x = 4 π k \frac{R^{3 + a}}{3 + a}$

$∴$ $Electric field at r = R$

$E_{1} = \frac{1}{4 π ε_{0}} \cdot \frac{4 π k R^{3 + a}}{(3 + a) R^{2}} = \frac{1}{4 π ε_{0}} \cdot \frac{4 π k}{3 + a} R^{1 + a}$

$For r = \frac{R}{2} :$

Total charge enclosed in the sphere of radius $\frac{R}{2}$

$Q' = \int_{0}^{R / 2} 4 π k x^{2 + a} d x = \frac{4 π k {(R / 2)}^{3 + a}}{3 + a}$

$∴$ $The electric field at r = \frac{R}{2}$

$E_{2} = \frac{1}{4 π ε_{0}} \cdot \frac{4 π k}{3 + a} \frac{{(R / 2)}^{3 + a}}{{(R / 2)}^{2}} = \frac{1}{4 π ε_{0}} \cdot \frac{4 π k}{3 + a} {(\frac{R}{2})}^{1 + a}$

$∵$ $E_{2} = \frac{1}{8} E_{1}$

$∴$ $\frac{1}{4 π ε_{0}} \cdot \frac{4 π k}{3 + a} {(\frac{R}{2})}^{1 + a} = \frac{1}{2^{3}} \times \frac{1}{4 π ε_{0}} \cdot \frac{4 π k}{3 + a} R^{1 + a}$

$\Rightarrow 1 + a = 3$

$∴$ $a = 2$

Solution

Q.
A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $ρ = k r^{a},$ where $k$ and $a$ are constants and $r$ is the distance from its centre. If the electric field at $r = \frac{R}{2}$ is $\frac{1}{8}$ times that at $r = R,$ find the value of $a$ . [2009]

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Solution

Q. A solid sphere of radius R has a charge Q distributed in its volume with a charge density ρ=kra, where k and a are constants and r is the distance from its centre. If the electric field at r=R2 is 18 times that at r=R, find the value of a. [2009]

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Q.
A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $ρ = k r^{a},$ where $k$ and $a$ are constants and $r$ is the distance from its centre. If the electric field at $r = \frac{R}{2}$ is $\frac{1}{8}$ times that at $r = R,$ find the value of $a$ . [2009]