To determine the value of $ K $ for the given electrical potential scenario, we start by calculating the potential at points C and D.

Potential at Point C:

The potential at point C, $ V_C $, is given by the sum of the potentials due to charges $ q_1 $ and $ q_2 $:

$ V_C = \frac{kq_1}{0.4} + \frac{kq_2}{0.5} $

Potential at Point D:

Similarly, the potential at point D, $ V_D $, is:

$ V_D = \frac{kq_1}{0.4} + \frac{kq_2}{0.1} $

Change in Potential Energy:

The change in potential energy $ \Delta U $ when $ q_3 $ moves from C to D is:

$ \Delta U = (V_D - V_C) \times q_3 $

Substituting the expressions for $ V_D $ and $ V_C $, we get:

$ \Delta U = \left(\frac{kq_2}{0.1} - \frac{kq_2}{0.5}\right) \times q_3 $

Simplifying the Expression:

By simplifying the expression, we find:

$ \Delta U = \left(\frac{5kq_2}{0.5}\right) \times q_3 - \left(\frac{kq_2}{0.5}\right) \times q_3 $

$ \Delta U = \frac{4kq_2}{0.5} \times q_3 = 8kq_2 q_3 $

$ \Delta U = \frac{8q_2 q_3}{4\pi \varepsilon_0} $

Thus, the value of $ K $ is $ 8q_2 $.