Electromagnetic Fields of a Moving Charge

Consider an electric charge, $q$, that is at rest at the origin of an inertial reference frame $S'$. The electric and magnetic fields generated by the charge at displacement ${\bf r}'$ in frame $S'$ are

$${\bf E}' =\frac{q}{4\pi\,\epsilon_0}\,\frac{{\bf r}'}{r'^{\,3}},$$ (3.286)

$${\bf B}' = {\bf0},$$ (3.287)

respectively. (See Sections 2.1.2 and 2.2.7.) Thus,

$$E_x' = \frac{q}{4\pi\,\epsilon_0}\,\frac{x'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}},$$ (3.288)

$$E_y' = \frac{q}{4\pi\,\epsilon_0}\,\frac{y'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}},$$ (3.289)

$$E_z' = \frac{q}{4\pi\,\epsilon_0}\,\frac{z'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}},$$ (3.290)

$$B_x' = B_y'=B_z'=0.$$ (3.291)

Let us transform to an inertial reference frame, $S$, that is in a standard configuration, and moves with velocity $-v\,{\bf e}_x$, with respect to frame $S'$. Thus, in frame $S$, the charge appears to move with velocity ${\bf v} = v\,{\bf e}_x$. Making use of the field transformation relations, (3.278)–(3.283), with primed and unprimed fields swapped, and $v\rightarrow -v$, we obtain

$$E_x = E_x' = \frac{q}{4\pi\,\epsilon_0}\,\frac{x'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}},$$ (3.292)

$$E_y = \gamma\,(E_y'+v\,B_z')= \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma\,y'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}},$$ (3.293)

$$E_z = \gamma\,(E_z'-v\,B_y') = \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma\,z'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}},$$ (3.294)

$$B_x = B_x' = 0,$$ (3.295)

$$B_y = \gamma\left(B_y' - \frac{v}{c^2}\,E_z'\right) = \frac{q}{4\pi\,... ...rac{\gamma\,z'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}}\left(-\frac{v}{c^2}\right),$$ (3.296)

$$B_z = \gamma\left(B_z' + \frac{v}{c^2}\,E_y'\right)= \frac{q}{4\pi\,\... ...rac{\gamma\,y'}{(x'^{\,2}+y'^{\,2}+z'^{\,2})^{3/2}}\left(+\frac{v}{c^2}\right).$$ (3.297)

Figure 3.18: Observing the field of a moving charge.

Consider the electric and magnetic fields generated by the charge at some point $P$ in frame $S$ whose displacement is ${\bf r} = (x,$ $y$, $z)$. See Figure 3.18. The displacement of the charge in frame $S$ is ${\bf r}' = (v\,t,$ 0, 0). Let

$${\bf s} = {\bf r} - {\bf r}'= (x-v\,t,~y,~z)$$ (3.298)

be a vector that is directed from the instantaneous position of the charge in frame $S$ to point $P$. A Lorentz transformation (see Section 3.2.7) between frames $S$ and $S'$ reveals that

$$x' = \gamma\,(x-v\,t) =\gamma s_x,$$ (3.299)

$$y' = y = s_y,$$ (3.300)

$$z' = z = s_z.$$ (3.301)

Let us write

$$s_x =s\,\cos\theta,$$ (3.302)

$$s_y = s\,\sin\theta\,\cos\varphi,$$ (3.303)

$$s_z =s\,\sin\theta\,\sin\varphi,$$ (3.304)

where $\theta$ is the angle subtended between ${\bf s}$ and ${\bf v}$, and $\varphi$ is an azimuthal angle. See Figure 3.18. It is easily demonstrated from the previous six equations that

$$x'^{\,2} + y'^{\,2} + z'^{\,2} = \gamma^2\,s_x^{\,2} + s_y^{\,2} + s_z^{\,2}$$

$$= \left(\gamma^2\,\cos^2\theta + \sin^2\theta\right)s^2=\left[(\gamma^2-1)\,\cos^2\theta+1\right]s^2$$

$$= \left(\frac{\gamma^2\,v_r^{\,2}}{c^2}+1\right)s^2,$$ (3.305)

where $v_r = v\,\cos\theta$ is the component of ${\bf v}$ that is directed from the instantaneous position of the charge to the point $P$. Thus, making use of Equations (3.292)–(3.297) and Equations (3.299)–(3.301), we obtain

$$E_x = \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma}{(1+\gamma^2\,v_r^{\,2}/c^2)^{3/2}}\,\frac{s_x}{s^3},$$ (3.306)

$$E_y = \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma}{(1+\gamma^2\,v_r^{\,2}/c^2)^{3/2}}\,\frac{s_y}{s^3},$$ (3.307)

$$E_z = \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma}{(1+\gamma^2\,v_r^{\,2}/c^2)^{3/2}}\,\frac{s_z}{s^3},$$ (3.308)

$$B_x =0,$$ (3.309)

$$B_y = \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma}{(1+\gamma^2\,v_r^{\,2}/c^2)^{3/2}}\,\frac{s_z}{s^3}\left(-\frac{v}{c^2}\right),$$ (3.310)

$$B_z = \frac{q}{4\pi\,\epsilon_0}\,\frac{\gamma}{(1+\gamma^2\,v_r^{\,2}/c^2)^{3/2}}\,\frac{s_y}{s^3}\left(+\frac{v}{c^2}\right).$$ (3.311)

Figure 3.19: Electric field-lines of a stationary (left) and a moving (right) electric charge.

Note that the electric field-lines generated by a moving electric charge are straight-lines that are directed from the instantaneous position of the charge to the point of observation. At low velocities (i.e., $v\ll c$), the field-lines are equally spaced around the charge. However, as $v\rightarrow c$, the field-lines become increasingly bunched in the plane transverse to the charge's direction of motion that passes through the charge. This is illustrated schematically in Figure 3.19. The magnetic field generated by a moving charge is

$${\bf B}= \frac{{\bf v}\times {\bf E}}{c^2},$$ (3.312)

where ${\bf v}$ is the charge's velocity, and ${\bf E}$ is the electric field generated by the charge.