📂Electrodynamics

Magnetic Field Created by a Moving Point Charge

Overview ¹

The electromagnetic field created by a moving point charge is as follows.

$$\begin{align*} \mathbf{E}(\mathbf{r}, t) &= \frac{q}{4\pi\epsilon_{0}} \frac{\cR} {( \bcR\cdot \mathbf{u} )^3 } \left[(c^2-v^2)\mathbf{u} +\bcR\times (\mathbf{u} \times \mathbf{a} ) \right] \\ \mathbf{B} (\mathbf{ r}, t) &=\frac{1}{c} \crH\times \mathbf{ E } (\mathbf{ r}, t) \end{align*}$$

Description

The formula for the magnetic field is specifically as follows.

$$\mathbf{B}=-\frac{1}{c}\frac{1}{4\pi \epsilon_{0}} \frac{q}{ (\mathbf{u}\cdot \bcR)^{3}} \bcR \times \left[ (c^{2}-v^{2})\mathbf{v}+(\bcR \cdot \mathbf{a})\mathbf{v}+(\bcR \cdot \mathbf{u})\mathbf{a} \right]$$

Introducing the derivation process for the magnetic field.

Derivation

The Liénard-Wiechert potential represents the potential created by a moving point charge.

$$V(\mathbf{r}, t)= \frac{1}{4\pi \epsilon_{0}} \frac{qc}{ (\cR c -\bcR\cdot \mathbf{v})} ,\quad \mathbf{A}(\mathbf{r}, t) = \frac{ \mathbf{v} } {c^2} V(\mathbf{r}, t)$$

The magnetic field is calculated as follows.

$$\quad \mathbf{B}=\nabla \times \mathbf{A}$$

Therefore,

$$\begin{align} \nabla \times \mathbf{A} &= \nabla \times \left( \frac{\mathbf{v}}{c^{2}}V(\mathbf{r},t) \right) \nonumber \\[1em] &= \frac{1}{c^{2}}\left(\nabla \times V \mathbf{v}\right) \nonumber \\[1em] &= \frac{1}{c^{2}} \Big( V(\nabla \times \mathbf{v})-\mathbf{v}\times(\nabla V) \Big) \end{align}$$

The third equals sign is validated by the multiplication rule for curl in $\nabla \times (f\mathbf{A}) = f(\nabla \times \mathbf{A}) - \mathbf{A} \times (\nabla f)$. The last two results, from $\nabla \times \mathbf{v}$, were already computed in the Electric Field Created by a Moving Point Charge.

$$\nabla \times \mathbf{v}=-\mathbf{a}\times \nabla t_{r}=\frac{\mathbf{a}\times \bcR}{\cR c - \bcR\cdot \mathbf{v}}$$

Therefore,

$$\begin{align*} V(\nabla \times \mathbf{v} )&= \frac{1}{4\pi \epsilon_{0}} \frac{qc}{ (\cR c -\bcR\cdot \mathbf{v})}\frac{\mathbf{a}\times \bcR}{\cR c - \bcR\cdot \mathbf{v}} \\ &= \frac{1}{4\pi \epsilon_{0}} \frac{qc(\mathbf{a}\times \bcR)}{ (\cR c -\bcR\cdot \mathbf{v})^{2}} \end{align*}$$

Here, if we set $\mathbf{u}=c\crH-\mathbf{v}$ to $\cR c - \bcR\cdot \mathbf{v}=\mathbf{u}\cdot \bcR$, and preliminarily convert it to an easier form to calculate,

$$\begin{equation} V(\nabla \times \mathbf{v} )= \frac{1}{4\pi \epsilon_{0}} \frac{qc(\mathbf{u}\cdot \bcR)(\mathbf{a}\times \bcR)}{ (\mathbf{u}\cdot \bcR)^{3}} \end{equation}$$

$\nabla V$ was also calculated in the same document.

$$\nabla V = \frac{qc}{4\pi\epsilon_{0}} \frac{1}{ (\cR c -\bcR \cdot \mathbf{v} )^3} \Big[ (\cR c -\bcR \cdot \mathbf{v})\mathbf{v} - (c^2 -v^2+\bcR \cdot \mathbf{a} ) \bcR \Big]$$

The cross product of the same vectors is $0$, so

$$\begin{align} \mathbf{v}\times \nabla V &= \frac{qc}{4\pi\epsilon_{0}} \frac{1}{ (\cR c -\bcR \cdot \mathbf{v} )^3}\mathbf{v}\times \Big[ - (c^2 -v^2+\bcR \cdot \mathbf{a} ) \bcR \Big] \nonumber \\ &= \frac{qc}{4\pi\epsilon_{0}} \frac{1}{ (\mathbf{u}\cdot \bcR)^3}\left[ -(c^{2}-v^{2})\mathbf{v}\times \bcR -(\bcR\cdot \mathbf{a})\mathbf{v}\times \bcR \right] \end{align}$$

Substituting $(2)$, $(3)$ into $(1)$ gives

$$\begin{align*} \mathbf{B}= \nabla \times \mathbf{A} &= \frac{1}{c^{2}}\frac{1}{4\pi \epsilon_{0}} \frac{qc}{ (\mathbf{u}\cdot \bcR)^{3}}\left[ (\mathbf{u}\cdot \bcR)(\mathbf{a}\times \bcR)+(c^{2}-v^{2})\mathbf{v}\times \bcR +(\bcR\cdot \mathbf{a})\mathbf{v}\times \bcR\right] \\ &=-\frac{1}{c}\frac{1}{4\pi \epsilon_{0}} \frac{q}{ (\mathbf{u}\cdot \bcR)^{3}} \bcR \times \left[ (c^{2}-v^{2})\mathbf{v}+(\bcR \cdot \mathbf{a})\mathbf{v}+(\bcR \cdot \mathbf{u})\mathbf{a} \right] \end{align*}$$

This result is quite similar to the form of the Electric Field Created by a Moving Point Charge.

$$\mathbf{E}(\mathbf{r},t) = \frac{q}{4\pi\epsilon_{0}}\frac{\cR}{(\bcR\cdot \mathbf{u})^{3}}\left[ (c^{2}-v^{2})\mathbf{u} + \mathbf{u}(\bcR\cdot\mathbf{a})-\mathbf{a}(\bcR\cdot\mathbf{u}) \right]$$

The only difference is that instead of $\mathbf{u}$, $\mathbf{v}$ is used. Indeed, since $\bcR\times \mathbf{u}=\cR \times (c\crH-\mathbf{v})=-\bcR\times \mathbf{v}$ holds, the following is obtained.

$$\begin{align*} \frac{1}{c}\crH\times \mathbf{E}(\mathbf{r},t) &= \frac{1}{c}\frac{1}{4\pi\epsilon_{0}}\frac{q}{(\bcR\cdot \mathbf{u})^{3}}\bcR\times \left[ (c^{2}-v^{2})\mathbf{u} + \mathbf{u}(\bcR\cdot\mathbf{a})-\mathbf{a}(\bcR\cdot\mathbf{u}) \right] \\ &=\frac{1}{c}\frac{1}{4\pi\epsilon_{0}}\frac{q}{(\bcR\cdot \mathbf{u})^{3}}\bcR\times \left[ (c^{2}-v^{2})(-\mathbf{v}) -\mathbf{v}(\bcR\cdot\mathbf{a})-\mathbf{a}(\bcR\cdot\mathbf{u}) \right] \\ &=-\frac{1}{c}\frac{1}{4\pi\epsilon_{0}}\frac{q}{(\bcR\cdot \mathbf{u})^{3}}\bcR\times \left[ (c^{2}-v^{2})\mathbf{v} + \mathbf{v}(\bcR\cdot\mathbf{a})+\mathbf{a}(\bcR\cdot\mathbf{u}) \right] \\ &=\mathbf{B}(\mathbf{r},t) \end{align*}$$

In other words, the magnetic field created by a point charge is orthogonal to the vector up to the delayed position with respect to the electric field.

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