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 can start by analyzing the forces acting on the charges and using Newton's laws of motion. Since the four positive charges are in motion, they experience a repulsive electrostatic force, while the negative charge at the center experiences an attractive electrostatic force.
By symmetry, the net force acting on each positive charge is directed towards the center of the square. This means that the motion of each charge is essentially one-dimensional, with only the distance from the center changing.
Let's consider one of the positive charges, labeled as charge k, at position r_k(t) at time t. The force acting on this charge can be derived from Coulomb's law:
F_k(t) = k * (q^2) / (|r_k(t)|^2)
Where k is the Coulomb constant and q is the charge of each particle.
Since we have a square configuration, the distance of each charge from the center is given by half the side length of the square:
|r_k(t)| = l(t) / √2
With l(t) being the side length of the square at time t.
Now, let's consider the equation of motion for a single positive charge:
m * d^2r_k(t) / dt^2 = F_k(t)
Substituting the expression for the force:
m * d^2r_k(t) / dt^2 = k * (q^2) / (l(t))^2 / 2
Simplifying the equation further:
d^2r_k(t) / dt^2 = (k * q^2) / (m * (l(t))^2) / 2
Since the motion is periodic with period T, we can assume a sinusoidal form for the displacement of the charges:
r_k(t) = A_k * cos(2πt / T + ϕ_k)
Where A_k is the amplitude and ϕ_k is the phase offset for charge k.
Taking the second derivative of r_k(t) with respect to t:
d^2r_k(t) / dt^2 = -A_k * (2π / T)^2 * cos(2πt / T + ϕ_k)
Substituting this into the equation of motion:
-A_k * (2π / T)^2 * cos(2πt / T + ϕ_k) = (k * q^2) / (m * (l(t))^2) / 2
We can now average this equation over one period T to eliminate the time-dependent terms. Since the motion is periodic, the average over one period of the cosine function is zero, resulting in:
0 = (k * q^2) / (m * (l_avg)^2) / 2
Where l_avg is the average side length of the square over one period T.
Next, we need to relate the average side length to the minimum and maximum side lengths, l_min and l_max, respectively. Since the side length oscillates between these values, the average can be given by:
l_avg = (l_min + l_max) / 2
Substituting this into the equation and solving for T:
0 = (k * q^2) / (m * ((l_min + l_max) / 2)^2) / 2
Simplifying the equation:
0 = 2 * (k * q^2) / (m * (l_min + l_max)^2)
Rearranging to solve for T:
T = 2 * π * √((m * (l_min + l_max)^2) / (2 * (k * q^2)))
We are given the relation (k * q^2) / (m * L_0^3) = 10^4 s^−2, so we can substitute this into the equation:
T = 2 * π * √((m * (l_min + l_max)^2) / (2 * 10^4 * L_0^3))
Now, given the values of m, q, and L_0, you can substitute them into the equation to find the value of T.