Calculate the work required to assemble three charged particles at...

The work required to assemble the system of charges is equal to the electrostatic potential energy of the system.

The potential energy $U$ of a system of three point charges is given by:
$U = \dfrac{1}{4\pi\epsilon_0} \left( \dfrac{q_1 q_2}{r_{12}} + \dfrac{q_2 q_3}{r_{23}} + \dfrac{q_3 q_1}{r_{31}} \right)$

Given:
$q_1 = 2.0 \times 10^{-5} \text{ C}$
$q_2 = 4.0 \times 10^{-5} \text{ C}$
$q_3 = 3.0 \times 10^{-5} \text{ C}$
$r_{12} = r_{23} = r_{31} = 10 \text{ cm} = 0.1 \text{ m}$
$\dfrac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$

Substituting the values:
$U = \dfrac{9 \times 10^9}{0.1} \left[ (2.0 \times 10^{-5})(4.0 \times 10^{-5}) + (4.0 \times 10^{-5})(3.0 \times 10^{-5}) + (3.0 \times 10^{-5})(2.0 \times 10^{-5}) \right]$

$U = 9 \times 10^{10} \left[ 8.0 \times 10^{-10} + 12.0 \times 10^{-10} + 6.0 \times 10^{-10} \right]$

$U = 9 \times 10^{10} \times (8.0 + 12.0 + 6.0) \times 10^{-10}$

$U = 9 \times 10^{10} \times 26 \times 10^{-10}$

$U = 9 \times 26$

$U = 234 \text{ J}$

Therefore, the work required to assemble the charges is $234 \text{ J}$.