Two identical point charges (q = +4.60 x 10-6 C) are fixed at opposite corners of a square whose sides have a length of 0.510 m. A test charge (q0 = -4.80 x 10-8 C), with a mass of 9.80 x 10-8 kg, is released from rest at one of the corners of the square. Determine the speed of the test charge when it reaches the center of the square.

To determine the speed of the test charge when it reaches the center of the square, we can use the principle of conservation of mechanical energy.

The potential energy of the test charge at the initial position is given by the formula:

Potential energy = (k * |q1*q2|) / r

Where:
k is the electrostatic constant (8.99 x 10^9 N m^2 / C^2)
q1 and q2 are the charges of the point charges (q1 = q2 = +4.60 x 10^-6 C)
r is the distance between the test charge and the fixed charges (r = 0.510 m)

The initial potential energy is then:

Potential energy initial = (8.99 x 10^9 N m^2 / C^2) * (4.60 x 10^-6 C)^2 / 0.510 m

Next, we can use the principle of conservation of mechanical energy to find the final kinetic energy of the test charge when it reaches the center of the square. At the center, the distance between the test charge and each fixed charge is:

Distance = (0.510 m) / √2

The final potential energy is given by:

Potential energy final = (8.99 x 10^9 N m^2 / C^2) * (4.60 x 10^-6 C)^2 / [(0.510 m) / √2]

Since the total mechanical energy is conserved, the initial potential energy is equal to the final mechanical energy:

Potential energy initial = Potential energy final + Kinetic energy

Rearranging the equation, we get:

Kinetic energy = Potential energy initial - Potential energy final

Now, we can calculate the values and plug them into the equation:

Potential energy initial = (8.99 x 10^9 N m^2 / C^2) * (4.60 x 10^-6 C)^2 / 0.510 m
Potential energy final = (8.99 x 10^9 N m^2 / C^2) * (4.60 x 10^-6 C)^2 / [(0.510 m) / √2]

Kinetic energy = Potential energy initial - Potential energy final

Finally, we can calculate the speed of the test charge by using the formula:

Kinetic energy = (1/2) * mass * velocity^2

Rearranging the equation, we get:

Velocity = √[(2 * Kinetic energy) / mass]

Plugging in the calculated kinetic energy and mass values will give us the speed of the test charge when it reaches the center of the square.