Four identical particles each have charge q and mass m. They are released from rest at the vertices
of a square of side L. How fast is each charge moving when their distance from the center of the
square doubles?
To solve this problem, we can use the principle of conservation of energy. Initially, when the particles are at the vertices of the square, the total energy of the system is zero since they are at rest. When the distance from the center of the square doubles, the particles will be at the midpoints of the sides of the square.
Let's assume that the distance from the center of the square to one of the particles is r initially and 2r when it reaches the midpoint of the side. We need to find the speed of the particles at this new position.
Step 1: Calculate the initial potential energy of the system.
The potential energy of each particle at the vertices of the square can be calculated as:
PE_initial = (1/4πε₀) * (q² / r)
The total potential energy of the system is the sum of the potential energies of each particle:
Total PE_initial = 4 * PE_initial = (4/4πε₀) * (q² / r)
Step 2: Calculate the final kinetic energy of the system.
As the particles reach the midpoint of the side, they will have only kinetic energy.
The total kinetic energy of the system is given by:
Total KE_final = 4 * (1/2) * m * v²
Step 3: Equate the initial potential energy to the final kinetic energy.
To find the speed of the particles at the new position, we set the initial potential energy equal to the final kinetic energy:
(4/4πε₀) * (q² / r) = 4 * (1/2) * m * v²
Simplifying, we get:
(1/4πε₀) * (q² / r) = m * v²
Step 4: Solve for the speed (v).
v² = (1/4πε₀) * (q² / (m * r))
Taking the square root of both sides, we get:
v = √((1/4πε₀) * (q² / (m * r)))
So, the speed of each charge when their distance from the center of the square doubles is given by:
v = √((1/4πε₀) * (q² / (m * 2r)))
To find the speed at which each charge is moving when their distance from the center of the square doubles, we need to analyze the forces acting on the charges.
Since all four charges are identical and have the same charge, they will repel each other due to the electrostatic force. This force follows Coulomb's Law, which states that the force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's assume that the distance from each charge to the center of the square is r when they are at rest. When their distance from the center doubles, their new distance becomes 2r.
At any given moment, each charge is repelled by the other three charges. The net force acting on each charge is the vector sum of the electrical forces due to the other three charges. Since the charges are at the vertices of a square, the symmetry of the system ensures that the net force acting on each charge is in the radial direction (towards or away from the center of the square).
The magnitude of the net force on each charge due to the other three charges can be calculated using Coulomb's Law. The force between two charges is given by:
F = k * (q1 * q2) / r^2
where F is the electrostatic force, k is the Coulomb's constant, q1 and q2 are the charges, and r is the distance between them.
Using this formula, we can calculate the magnitude of the net force acting on each charge due to the other three charges.
Next, we can apply Newton's second law, which states that the net force acting on an object is equal to its mass times its acceleration:
Fnet = m * a
Since the net force is acting in the radial direction, the resulting acceleration of each charge will also be in that direction.
Now, we can solve for the acceleration of each charge by dividing the net force by its mass:
a = Fnet / m
Once we have the acceleration, we can use kinematic equations to find the final velocity of each charge. Assuming the charges start from rest, the equation that relates final velocity (v), initial velocity (u), acceleration (a), and displacement (s) is:
v^2 = u^2 + 2as
Since the charges start from rest, the initial velocity (u) is zero. The displacement (s) is the change in their distance from the center, which is 2r - r = r.
Simplifying the equation, we get:
v = sqrt(2 * a * r)
Finally, substitute the calculated acceleration (a) and the initial distance (r) to find the speed at which each charge is moving when their distance from the center of the square doubles.
Repeat the above calculations for each charge, and you will find the speed at which each charge is moving.
Calculate the potential energy of a particle at original location
( in general at distance integral from infinity to d of k Q1Q2 dr /r^2
= k Q1Q2 /r )
Do it again for bigger distance.
(1/2) m v^2 = decrease in potential energy