Electric field
Arbitrarily moving point charge

For arbitrarily moving point charges, propagation of potential fields such as Lorenz gauge fields at the speed of light needs to be accounted for by using Liénard–Wiechert potential. Since the potentials satisfy Maxwell's equations, the fields derived for point charge also satisfy Maxwell's equations. The electric field is expressed as: E (r, t) = 1 4 π ε 0 (q (n s − β s) γ 2 (1 − n s ⋅ β s) 3 | r − r s | 2 + q n s × ((n s − β s) × β s ˙) c (1 − n s ⋅ β s) 3 | r − r s |) t = t r {\displaystyle \mathbf {E} (\mathbf {r},\mathbf {t})={\frac {1}{4\pi \varepsilon _{0}}}\left({\frac {q(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big)}}{c(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}} where q {\displaystyle q} is the charge of the point source, t r {\textstyle {t_{r}}} is retarded time or the time at which the source's contribution of the electric field originated, r s (t) {\textstyle {r}_{s}(t)} is the position vector of the particle, n s (r, t) {\textstyle {n}_{s}(\mathbf {r},t)} is a unit vector pointing from charged particle to the point in space, β s (t) {\textstyle {\boldsymbol {\beta }}_{s}(t)} is the velocity of the particle divided by the speed of light, and γ (t) {\textstyle \gamma (t)} is the corresponding Lorentz factor. The retarded time is given as solution of: t r = t − | r − r s (t r) | c {\displaystyle t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}}
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