This example finds the average radiation power of two attracting charges moving in an elliptical orbit (an electric dipole).
The two opposite charges,
e2, form an electric dipole. The masses of the charged particles are
m2, respectively. For the common center of mass
m1*r1 + m2*r2 = 0, where
r2 are distance vectors to the charged particles. The distance between charged particles is
r = r1 - r2.
syms m1 m2 e1 e2 r1 r2 r [r1,r2] = solve(m1*r1 + m2*r2 == 0, r == r1 - r2, r1, r2)
Find the dipole moment of this system:
d = e1*r1 + e2*r2; simplify(d)
According to the Larmor formula, the total power radiated in a unit of time is , or, in terms of the distance between the charged particles, . Here dot means a time derivative. Coulomb's law lets you find the values of acceleration in terms of the reduced mass of the system, , and the product of the charges of the particles, .
alpha = sym('alpha'); syms m c m = m1*m2/(m1 + m2); r2 = -alpha/(m*r^2); J = simplify(subs(2/(3*c^3)*d^2, r, r2))
The major semiaxis a and eccentricity of an elliptical orbit are given by the following expressions, where
E is the total orbital energy, and is the angular momentum.
syms E L phi a = alpha/(2*E)
eccentricity = sqrt(1-2*E*L^2/(m*alpha^2))
The equation of an elliptical orbit, , lets you express the distance
r in terms of the angle
r = a*(1 - eccentricity^2)/(1 + eccentricity*cos(phi));
The average radiation power of two charged particles moving in an elliptical orbit is an integral of the radiation power over one full cycle of motion, normalized by the period of motion, . The period of motions
T = 2*pi*sqrt(m*a^3/alpha);
Changing the integration variable
phi, you get the following result. Use the
simplify function to get a shorter integration result. Here, use
subs to evaluate
J = subs(J); Javg = simplify(1/T*int(J*m*r^2/L, phi, 0, 2*pi))
Estimate the average radiation power of the electric dipole with one particle much heavier than the over,
m1>>m2. For this, compute the limit of the expression for radiation power, assuming that
m1 tends to infinity.
limJ = limit(Javg, m1, Inf); simplify(limJ)