I Derivative of the retarded vector potential

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The discussion focuses on deriving the time derivative of the scalar potential V in the context of an oscillating electric dipole. The potential vector A is expressed in terms of the dipole moment p and its time derivative. Using the Lorentz Gauge condition, the relationship between A and V is established, leading to the need for calculating the derivative of V. The final result for the time derivative of V incorporates both the second and first time derivatives of the dipole moment, highlighting the influence of the dipole's motion on the potential. The conversation emphasizes the importance of applying the chain rule in calculus for this derivation.
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In a problem of an oscillating electric dipole, under appropriate conditions, one can find, for the potential vector calculated at the point ##\vec{r}##, the expression ##\vec{A}=\hat{k}\frac{\mu_0I_0d}{4\pi}\frac{cos(\omega(t-r/c))}{r}## where: ##\hat{k}## is the direction of the ##z-axis## where the dipole is oscillating, ##I_0## is the current (##I(t)=I_0cos(\omega t)##), ##d## is the distance between the charges of the dipole and ##r## is the distance between the origin of the system and the point where I want to calculate the potential vector. Let ##\vec{p}=\hat{k}qd=\frac{\hat{k}dI_0}{\omega}sin(\omega t)## be the dipole moment, it is possible to rewrite the potential vector as ##\vec{A}=\frac{\mu_0}{4\pi}\frac{\vec{\dot p(t-r/c)}}{r}## where ##\vec{\dot p}## is the derivative with respect to time.
If we use Lorentz Gauge we have ##-\epsilon_0\mu_0\frac{\partial V}{\partial t}=\vec{\nabla} \cdot \vec{A}##, since ##\vec{A}## is directed along the ##z-axis## we have ##div \vec{A}=\frac{\partial A}{\partial z}## so ##\frac{\partial V}{\partial t}=-\frac{1}{\epsilon_0 \mu_0}\frac{\partial A}{\partial z}##.

The question: how do you do this last derivative?

The result is: ##\frac{\partial V}{\partial t}=\frac{1}{4\pi\epsilon_0} \left( \frac{- \ddot p(t-r/c)}{cr}+\frac{\dot p(t-r/c)}{r^2} \right)\frac{z}{r}##
 
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