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Electric Field Due to a Sphere and Thin Shell

Question

Surface density of charge on a sphere of radius $R$ in terms of electricity intensity $E$ at a distance $r$ in free space is ________.
( $ϵ_{0}$ = permittivity of free space)

$ϵ_{0} E {(\frac{R}{r})}^{2}$
$\frac{ϵ_{0} E R}{r^{2}}$
$ϵ_{0} E {(\frac{r}{R})}^{2}$
$\frac{ϵ_{0} E r}{R^{2}}$

A

ϵ_{0} E {(\frac{R}{r})}^{2}

B

ϵ_{0} E {(\frac{r}{R})}^{2}

C

\frac{ϵ_{0} E R}{r^{2}}

D

\frac{ϵ_{0} E r}{R^{2}}

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Solution

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At distances greater than the radius from the center, the electric field due to a charged sphere is same as that of a point charge of same magnitude:

$E = \frac{Q}{4 π ϵ_{0} r^{2}}$

Now, total charge and surface charge density are related by:

$Q = 4 π R^{2} σ$

This gives:

$σ = ϵ_{0} E (\frac{r}{R})^{2}$

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