A conducting sphere of radius 10 cm has an unknown charge.
Question:

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?

Solution:

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.