Check that the potentials of a point charge moving at constant velocity (Eqs. 10.49 and 10.50) satisfy the Lorenz gauge condition (Eq. 10.12)

Question

[latex]V( r , t)=\frac{1}{4 \pi \epsilon_{0}} \frac{q c}{\sqrt{\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight)}}[/latex] (10.49)

[latex]A ( r , t)=\frac{\mu_{0}}{4 \pi} \frac{q c v }{\sqrt{\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight)}}[/latex] (10.50)

[latex] abla \cdot A =-\mu_{0} \epsilon_{0} \frac{\partial V}{\partial t}[/latex] (10.12)

Accepted Answer

Using Product Rule #5, Eq. 10.50 ⇒

[latex] abla \cdot A =\frac{\mu_{0}}{4 \pi} q c v \cdot abla\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-1 / 2}[/latex]

[latex]=\frac{\mu_{0} q c}{4 \pi} v \cdot\left\{-\frac{1}{2}\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-3 / 2} abla \left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight] ight\}[/latex]

[latex]=-\frac{\mu_{0} q c}{8 \pi}\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-3 / 2} v \cdot\left\{-2\left(c^{2} t- r \cdot v ight) abla ( r \cdot v )+\left(c^{2}-v^{2} ight) abla \left(r^{2} ight) ight\}[/latex].

Product Rule #4 ⇒

[latex] abla ( r \cdot v )= v imes( abla imes r )+( v \cdot abla ) r , ext { but } abla imes r =0[/latex],

[latex]( v \cdot abla ) r =\left(v_{x} \frac{\partial}{\partial x}+v_{y} \frac{\partial}{\partial y}+v_{z} \frac{\partial}{\partial z} ight)(x \hat{ x }+y \hat{ y }+z \hat{ z })=v_{x} \hat{ x }+v_{y} \hat{ y }+v_{z} \hat{ z }= v[/latex] , and

[latex] abla \left(r^{2} ight)= abla ( r \cdot r )=2 r imes( abla imes r )+2( r \cdot abla ) r =2 r[/latex] . So

[latex] abla \cdot A =-\frac{\mu_{0} q c}{8 \pi}\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-3 / 2} v \cdot\left[-2\left(c^{2} t- r \cdot v ight) v +\left(c^{2}-v^{2} ight) 2 r ight][/latex]

[latex]=\frac{\mu_{0} q c}{4 \pi}\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-3 / 2}\left\{\left(c^{2} t- r \cdot v ight) v^{2}-\left(c^{2}-v^{2} ight)( r \cdot v ) ight\}[/latex].

But the term in curly brackets is : [latex]c^{2} t v^{2}-v^{2}( r \cdot v )-c^{2}( r \cdot v )+v^{2}( r \cdot v )=c^{2}\left(v^{2} t- r \cdot v ight)[/latex].

[latex]=\frac{\mu_{0} q c^{3}}{4 \pi} \frac{\left(v^{2} t- r \cdot v ight)}{\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{3 / 2}}[/latex].

Meanwhile, from Eq. 10.49,

[latex]-\mu_{0} \epsilon_{0} \frac{\partial V}{\partial t}=-\mu_{0} \epsilon_{0} \frac{1}{4 \pi \epsilon_{0}} q c\left(-\frac{1}{2} ight)\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-3 / 2} imes[/latex]

[latex]\frac{\partial}{\partial t}\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight][/latex]

[latex]=-\frac{\mu_{0} q c}{8 \pi}\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{-3 / 2}\left[2\left(c^{2} t- r \cdot v ight) c^{2}+\left(c^{2}-v^{2} ight)\left(-2 c^{2} t ight) ight][/latex]

[latex]=-\frac{\mu_{0} q c^{3}}{4 \pi} \frac{\left(c^{2} t- r \cdot v -c^{2} t+v^{2} t ight)}{\left[\left(c^{2} t- r \cdot v ight)^{2}+\left(c^{2}-v^{2} ight)\left(r^{2}-c^{2} t^{2} ight) ight]^{3 / 2}}= abla \cdot A[/latex].

Question 10.27: Check that the potentials of a point charge moving at consta...

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