Four charges are fixed at the corners of a square centered at the origin as follows: charge q at (-a, +a), charge 2q at (+a, +a), charge -3q at (+a, -a), and charge 6q at (-a, -a). A fifth charge, +q with mass m, is placed at the origin and released from rest. Determine the speed of the fifth charge at a great distance from the origin, where the potential energy due to the four point charges is negligible.

To determine the speed of the fifth charge +q at a great distance from the origin, we can use the principle of conservation of energy. Since there are no non-conservative forces doing work, the total mechanical energy of the system (kinetic energy + potential energy) remains constant.

Initially, when the charge is at the origin, its kinetic energy (K.E.) is zero because it's released from rest. The electric potential energy (U) is due to the interaction with the four fixed charges at the corners of the square.

Step 1: Calculate the initial potential energy of charge +q due to the four other charges using Coulomb's Law. For point charges, the potential energy (U) between two charges q1 and q2 that are r distance apart is:

U = (k * q1 * q2) / r

Where k is Coulomb's constant (8.99 x 10^9 N·m²/C²). We would calculate this for each pair of charges (+q with each of the four fixed charges) and sum these up for the total potential energy when the charge is at the origin.

Step 2: At a very large distance, the potential energy due to these point charges will be negligible. Now the charge +q will only have kinetic energy, and it will be equal to the initial potential energy because of energy conservation:

(K.E.)_final = (U)_initial

Since kinetic energy (K.E.) is given by (1/2)mv^2, and we know the (U)_initial from Step 1, solve for the final speed v:

(1/2)m*v^2 = (U)_initial

v = sqrt[(2*(U)_initial)/m]

Step 3: Replace (U)_initial with its calculated value from Coulomb’s law in the previous step, and solve for v.

Using this approach, you can find the speed of the charge +q as it moves away from the origin under the influence of the other four charges.

Extra: Coulomb's Law is a fundamental principle in electromagnetism that describes the force between two point charges. The force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them:

F = k * |q1 * q2| / r^2

The concept of potential energy in electrostatics is that a charge in an electric field has the potential to do work due to its position. As the charge moves away from the other charges, it can either gain or lose potential energy depending on whether it’s moving against or along the electric field, respectively. The potential energy plays a crucial role when dealing with forces and energies in electrostatics. Also, note that the potential at a great distance from a group of charges tends to zero (which is why we've assumed the potential energy is negligible at a great distance for this problem).

The conservation of energy is a principle stating that energy cannot be created or destroyed in an isolated system. The energy can change forms, for example, potential energy can become kinetic energy and vice versa, but the total amount of energy remains constant. This principle allows us to equate the initial potential energy with the final kinetic energy to find the unknown, which, in this case, is the final speed of the charge.