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(B) The electric field $E$ at a distance $r$ from the center of a charged conducting sphere of radius $R$ is given by Gauss's Law as $E = \frac{1}{4 \

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$\varepsilon_{0} E \left(\frac{r}{R}\right)^{2}$

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(B) The electric field $E$ at a distance $r$ from the center of a charged conducting sphere of radius $R$ is given by Gauss's Law as $E = \frac{1}{4 \pi \varepsilon_{0}} \frac{Q}{r^{2}}$.Since the total charge $Q$ on the sphere is related to the surface charge density $\sigma$ by $Q = \sigma \cdot (4 \pi R^{2})$,we substitute this into the electric field equation.$E = \frac{1}{4 \pi \varepsilon_{0}} \frac{\sigma (4 \pi R^{2})}{r^{2}}$.Simplifying the expression,we get $E = \frac{\sigma R^{2}}{\varepsilon_{0} r^{2}}$.Rearranging the formula to solve for $\sigma$,we get $\sigma = E \varepsilon_{0} \left(\frac{r}{R}\right)^{2}$.

The surface charge density $\sigma$ on a charged conducting sphere of radius $R$ in terms of the electric field intensity $E$ at a distance $r$ $(r > R)$ in free space is (where $\varepsilon_{0}$ is the permittivity of free space):

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