The torque τ experienced by an electric dipole in an electric field is given by the formula:

$$\tau = pE\sin{\theta}$$

where p is the electric dipole moment, E is the electric field intensity, and θ is the angle between the dipole and the electric field. The electric dipole moment p can be expressed as:

$p = qd$

where q is the charge on the dipole, and d is the dipole length.

We are given the following values:

• Torque τ = 4 N·m
• Electric field intensity E = $$2 \times 10^5 ~\mathrm{NC}^{-1}$$
• Angle θ = $30^{\circ}$
• Dipole length d = 2 cm = 0.02 m

We need to find the charge q on the dipole. Let's first solve for the electric dipole moment p:

$$\tau = pE\sin{\theta}$$

$ \Rightarrow $ $$p = \frac{\tau}{E\sin{\theta}}$$

Substituting the given values:

$$p = \frac{4}{(2 \times 10^5) \sin{30^{\circ}}} = \frac{4}{(2 \times 10^5)(0.5)} = \frac{4}{10^5} = 4 \times 10^{-5} ~\mathrm{C~m}$$

Now, let's solve for the charge q using the formula:

$ \Rightarrow $ $p = qd$

$q = \frac{p}{d}$

Substituting the values for p and d:

$$q = \frac{4 \times 10^{-5}}{0.02} = 2 \times 10^{-3} ~\mathrm{C} = 2 ~\mathrm{mC}$$

So, the magnitude of the charge on the dipole is 2 mC.