Exact solutions for many-body problems are rarely encountered in physics. The following problem deals with a non-trivial motion of four charges. Due to the symmetry of the problem it is possible to determine the trajectories of the charges analytically.
Four identical particles with mass m and charge +q, orbit a charge −q as shown in the figure. The four positive charges always form a square of side l(t) while the negative charge stays at rest at the center of the square. The motion of the charges is periodic with period T. That is, if the vectors r_k(t), k={1,2,3,4}, describe the position of the charges then we have that
r_k(t+T)=r_k(t).
It is also known that the side of the square oscillates between l_min=1/4L_0 and l_max=L_0. Determine the period T in seconds if the parameters q,m and L_0 satisfy the relation
(kq^2)/(m(L_0^3))=10^4 s^−2,where k=1/(4πϵ_0)
To determine the period T, we need to find the relationship between T and the parameters q, m, and L_0 using the given relation. However, before proceeding, let's understand the problem setup and key concepts related to the motion of charges.
The problem describes four identical particles with mass m and charge +q that orbit a central charge -q. The positive charges always form a square with side length l(t), while the negative charge remains stationary at the center of the square. The motion of the charges is periodic, meaning that after a certain time interval T, the charges return to their initial positions.
To solve this problem analytically, we can use the principles of classical mechanics and the laws of electrostatics. We need to find the equation of motion for the charges and then determine the relationship between T and the given parameters.
Let's break down the problem into steps:
Step 1: Find the equation of motion for the charges.
To analyze the motion of the particles, we can apply Newton's second law for each charge. The forces acting on each particle are due to the electrostatic interaction between them. Since the positive charges form a square with side length l(t), we can consider the forces acting between adjacent pairs of charges and sum them up.
Considering the symmetry of the system, we can assume that the charges move in the x-y plane, and their positions can be described using vectors r_k(t), where k = {1,2,3,4} denotes the index of the charges.
Assuming small oscillations around the equilibrium positions, we can expand the forces in terms of small displacements from equilibrium. Then we can derive the equation of motion for each charge using Newton's second law.
Step 2: Solve the equation of motion.
Once we have the equation of motion, we can solve it for the variable l(t) and express it as a function of time. The specific form of the equation depends on the assumptions and simplifications made in Step 1.
Step 3: Analyze the periodic behavior.
By solving the equation of motion, we should obtain a time-dependent solution for l(t). Since the motion is periodic, l(t) will repeat itself after a period T. We can identify the minimum and maximum values of l(t) from the given information in the problem: l_min = 1/4L_0 and l_max = L_0.
Step 4: Determine the relationship between T and the parameters.
Using the time-dependent solution for l(t), we can find the relationship between T and the parameters q, m, and L_0. To do this, we need to express l(t) in terms of specific mathematical functions or find a way to calculate the time for one complete period of oscillation.
Finally, we can substitute the given relation into the equation and solve for T in seconds.
Note that the steps described above provide a general approach to finding the period T analytically. The specific details and mathematical calculations can vary depending on the assumptions and approximations made in the problem setup.