Introduction to Electrodynamics · Problem 10.34

Potentials and Fields — David J. Griffiths

Physics · retarded-potentials

Problem

Find the (Lorenz gauge) potentials and fields of a time-dependent ideal electric dipole p(t)\mathbf{p}(t) at the origin. (It is stationary, but its magnitude and/or direction are changing with time.) Don't bother with the contact term. [Answer:[Answer: V(r,t)=14πϵ0r^r2⋅[p+(r/c)p˙]V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0}\frac{\hat{\mathbf{r}}}{r^2}\cdot[\mathbf{p} + (r/c)\dot{\mathbf{p}}] A(r,t)=μ04π[p˙r]\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\left[\frac{\dot{\mathbf{p}}}{r}\right] E(r,t)=−μ04π{p¨−r^(r^⋅p¨)r+c2[p+(r/c)p˙]−3r^(r^⋅[p+(r/c)p˙])r3}\mathbf{E}(\mathbf{r},t) = -\frac{\mu_0}{4\pi}\left\{\frac{\ddot{\mathbf{p}} - \hat{\mathbf{r}}(\hat{\mathbf{r}}\cdot\ddot{\mathbf{p}})}{r} + c^2\frac{[\mathbf{p}+(r/c)\dot{\mathbf{p}}] - 3\hat{\mathbf{r}}(\hat{\mathbf{r}}\cdot[\mathbf{p}+(r/c)\dot{\mathbf{p}}])}{r^3}\right\} B(r,t)=−μ04π{r^×[p˙+(r/c)p¨]r2}\mathbf{B}(\mathbf{r},t) = -\frac{\mu_0}{4\pi}\left\{\frac{\hat{\mathbf{r}}\times[\dot{\mathbf{p}}+(r/c)\ddot{\mathbf{p}}]}{r^2}\right\} where all the derivatives of p\mathbf{p} are evaluated at the retarded time.]

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