Resonant motion and kinetic energy of charged particles in a time-dependent electromagnetic field

Problem Statement

Consider a charged particle of mass $m$m and electric charge $q$q ($q > 0$q>0) moving parallel to the $xy$xy plane in an infinitely expanding vacuum. A constant, uniform magnetic field $\boldsymbol{B} = (0, 0, B)$B=(0,0,B) ($B > 0$B>0) is applied in the positive direction of the $z$z axis. Furthermore, a time-varying, uniform electric field $\boldsymbol{E} = (0, E_0 \sin(\omega t), 0)$E=(0,E0​sin(ωt),0) ($E_0 > 0$E0​>0) is applied in the direction of the $y$y axis at time $t \ge 0$t≥0. Here, the angular frequency $\omega$ω is exactly equal to the particle's cyclotron angular frequency $\omega_c = \frac{qB}{m}$ωc​=mqB​, and the particle is assumed to be in a resonant state.

At time $t = 0$t=0, the charged particle was at rest at the origin $(0,0,0)$(0,0,0). Let the particle's velocity vector be $\boldsymbol{v}(t) = (v_x(t), v_y(t), 0)$v(t)=(vx​(t),vy​(t),0). Develop the equation of motion and find the time evolution of each velocity component. Furthermore, find the kinetic energy $K$K of this charged particle at time $t = \frac{10\pi}{\omega_c}$t=ωc​10π​, which is the time elapsed from time $t = 0$t=0, corresponding to 5 periods of cyclotron motion.

Assume that electromagnetic radiation from the particle and relativistic effects are all negligible.

Constraints

• $m = 5.00 \times 10^{-4} \text{ kg}$m=5.00×10−4 kg
• $q = 3.00 \times 10^{-2} \text{ C}$q=3.00×10−2 C
• $B = 0.500 \text{ T}$B=0.500 T
• $E_0 = 600 \text{ V/m}$E0​=600 V/m

Input Format

The kinetic energy $K \text{ [J]}$K [J] of a particle at time $t = \frac{10\pi}{\omega_c}$t=ωc​10π​ is expressed as $K = A \pi^2$K=Aπ2 using pi $\pi$π. Find the value of the coefficient $A$A and input a natural number representing that value.