A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of sphere is $1.5 \times 10^{3}\,\text{N C}^{-1}$ and points radially inward, what is the net charge on the sphere?

Solution:

Given, radius of sphere $R = 10 \,\text{cm} = 0.10 \,\text{m}$
Distance from centre, $r = 20 \,\text{cm} = 0.20 \,\text{m}$
Electric field at distance $r$ from centre, $E = 1.5 \times 10^{3} \,\text{N C}^{-1}$

The electric field due to charged sphere at external point distance $r$ from centre is

$$
E = \frac{1}{4\pi \epsilon_0} \cdot \frac{q}{r^2}
$$

Substituting the given values,

$$
1.5 \times 10^{3} = 9 \times 10^{9} \cdot \frac{q}{(0.20)^2}
$$

$$
\Rightarrow \, q = \frac{1.5 \times 10^{3} \times (0.20)^2}{9 \times 10^{9}}
= 6.67 \times 10^{-9} \,\text{C} = 6.67 \,\text{nC}
$$

As electric field is radially inward, charge on sphere is negative.
Therefore, charge on sphere

$$
q = -6.67 \,\text{nC}
$$