An electric dipole of length 2 cm is placed with its axis making an angle of $60^{\circ}$ to a uniform…

The torque on an electric dipole in a uniform electric field is given by $\tau = pE \sin\theta$, where $\tau$ is the torque, $p$ is the electric dipole moment, $E$ is the electric field magnitude, and $\theta$ is the angle between the dipole moment and field vectors.

Using the given values $E = 10^{5}\,\mathrm{N/C}$, $2a = 0.02\,\mathrm{m}$, $\theta = 60^\circ$, and $\tau = 9\sqrt{3}\,\mathrm{N\cdot m}$, we substitute into the torque equation:
$9\sqrt{3} = p \cdot 10^{5} \cdot \frac{\sqrt{3}}{2}$

Dividing both sides by $\sqrt{3}$:
$9 = p \cdot 10^{5} \cdot \frac{1}{2}$

Solving for $p$:
$p = \frac{18}{10^{5}} = 18 \times 10^{-5}\ \mathrm{C\cdot m}$

The electric dipole moment relates to charge by $p = q \cdot 2a$. Substituting $p$ and $2a$:
$18 \times 10^{-5} = q \cdot 0.02$

Solving for $q$:
$q = \frac{18 \times 10^{-5}}{2 \times 10^{-2}} = 9 \times 10^{-3}\ \mathrm{C}$

The magnitude of the charge is $9\,\mathrm{mC}$, corresponding to option C.