A conducting sphere of radius $10 \mathrm{~cm}$ has an unknown charge. If the electric field $20 \mathrm{~cm}$ from the centre of the sphere is $1.5 \times 10^{3} \mathrm{~N} / \mathrm{C}$ and points radially inward, what is the net charge on the sphere?
Electric field intensity (E) at a distance (d) from the centre of a sphere containing net charge q is given by the relation,
$E=\frac{q}{4 \pi \in_{0} d^{2}}$
Where,
$q=$ Net charge $=1.5 \times 10^{3} \mathrm{~N} / \mathrm{C}$
$d=$ Distance from the centre $=20 \mathrm{~cm}=0.2 \mathrm{~m}$
$\epsilon_{0}=$ Permittivity of free space
And $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{~N} \mathrm{~m}^{2} \mathrm{C}^{-2}$
$\therefore q=E\left(4 \pi \epsilon_{0}\right) d^{2}$
$=\frac{1.5 \times 10^{3} \times(0.2)^{2}}{9 \times 10^{9}}$
$=6.67 \times 10^{-9} \mathrm{C}$
= 6.67 nC
Therefore, the net charge on the sphere is 6.67 nC.
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