- #1
Frostman
- 115
- 17
- Homework Statement
- In an inertial reference system, a particle of mass ##m## and charge ##q## is given, with initial speed ##v(0) = (v_x (0); v_y (0); v_z (0))##. Furthermore, there is an electric field ##E##, parallel to the y-axis, and a magnetic field ##B##, parallel to the z-axis, both constant, homogeneous and such that ##|E| = |B|## in natural units.
Calculate the trend of the four-momenta ##p^\mu## as a function of the proper time ##\tau## and of the initial speed.
- Relevant Equations
- ##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu##
As a starting point I immediately thought about the equation:
##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu##
From this I proceed component by component:
##\frac{dp^0}{d\tau}=qF^{0\nu}v_\nu=q\gamma E_yv_y##
##\frac{dp^1}{d\tau}=qF^{1\nu}v_\nu=q\gamma v_yB_z##
##\frac{dp^2}{d\tau}=qF^{2\nu}v_\nu=q\gamma (E_y-u_xB_z)##
##\frac{dp^3}{d\tau}=qF^{3\nu}v_\nu=0##
Now that I have the values I integrate:
##p^0=\int_{p^0(0)}^{p^0(\tau)}dp^0=\int_{0}^{\tau}d\tau q\gamma E_yv_y##
##p^1=\int_{p^1(0)}^{p^1(\tau)}dp^0=\int_{0}^{\tau}d\tau q\gamma v_yB_z##
##p^2=\int_{p^2(0)}^{p^2(\tau)}dp^0=\int_{0}^{\tau}d\tau q\gamma (E_y-u_xB_z)##
##p^3=\int_{p^3(0)}^{p^3(\tau)}dp^0=\int_{0}^{\tau}d\tau 0=0##
I have a problem solving integrals with respect to ##\tau##, maybe I solved the last component, but I'm not sure:
##p^3(\tau) - p^3(0) = 0 \rightarrow p^3(\tau)= \text{cost} = m\gamma v_z(0)##
Can you tell me first of all if everything I have done is correct and if it remains for me to understand how to solve the integrals? If so, how can I proceed in solving these integrals? Is it okay that I analyze component by component or should I write it all in a formula like ##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu##?
##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu##
From this I proceed component by component:
##\frac{dp^0}{d\tau}=qF^{0\nu}v_\nu=q\gamma E_yv_y##
##\frac{dp^1}{d\tau}=qF^{1\nu}v_\nu=q\gamma v_yB_z##
##\frac{dp^2}{d\tau}=qF^{2\nu}v_\nu=q\gamma (E_y-u_xB_z)##
##\frac{dp^3}{d\tau}=qF^{3\nu}v_\nu=0##
Now that I have the values I integrate:
##p^0=\int_{p^0(0)}^{p^0(\tau)}dp^0=\int_{0}^{\tau}d\tau q\gamma E_yv_y##
##p^1=\int_{p^1(0)}^{p^1(\tau)}dp^0=\int_{0}^{\tau}d\tau q\gamma v_yB_z##
##p^2=\int_{p^2(0)}^{p^2(\tau)}dp^0=\int_{0}^{\tau}d\tau q\gamma (E_y-u_xB_z)##
##p^3=\int_{p^3(0)}^{p^3(\tau)}dp^0=\int_{0}^{\tau}d\tau 0=0##
I have a problem solving integrals with respect to ##\tau##, maybe I solved the last component, but I'm not sure:
##p^3(\tau) - p^3(0) = 0 \rightarrow p^3(\tau)= \text{cost} = m\gamma v_z(0)##
Can you tell me first of all if everything I have done is correct and if it remains for me to understand how to solve the integrals? If so, how can I proceed in solving these integrals? Is it okay that I analyze component by component or should I write it all in a formula like ##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu##?